H.     F\     LOOMIS, 

OEKKKAL   AliENT  FOR 

Architectural,  Mechanical, 

i  Buiimi  toon 

21  William  Street, 
WORCESTER,    MASS. 


Goddard  &  Nyc,  Pr».,  Worcette 


THE 


PRINCIPLES  OF  MECHANISM 


MACHINERY  OF  TRANSMISSION. 


COMPRISING 


THE   PRINCIPLES    OF   MECHANISM,   WHEELS    AND    PULLEYS, 

STRENGTH  AND  PROPORTIONS  OF  SHAFTS,  COUPLINGS  FOR 

SHAFTS,  AND  ENGAGING  AND  DISENGAGING  GEAR. 


BY 

WILLIAM  FAIRBAIRN,  ESQ.,  C.E. 

LL.D.   F.R.S.   F.G.S. 

COKKIirl'O.N-IU.VU  MEMBER  OF  TUB  NATIONAL  INSTITUTE  OP  FRANCE,  AND  OF  THB 
ROYAL  ACADEMY  OF  TURIN;  CHEVALIER  OF  THE 
LEUION  OP  HONOUR,  ETC.,  ETC. 


PHILADELPHIA: 

HENRY     CAREY     BAIRD, 

INDUSTRIAL     PUBLISHER, 
No.  406  WALNUT  STREET. 

1869. 

*,!»• 


PUBLISHER'S   PREFACE. 

This  work,  now  offered  to  the  public,  on 
the  "Principles  of  Mechanism  and  Trans- 
mission," is  taken  from  the  work  of  the 
distinguished  author  on  "Mills  and  Mill- 
work,"  which  is  large  and  expensive,  and 
contains  many  details  of  Millwork  not 
entirely  adapted  to  American  practice.  The 
great  principles  here  given  lie  at  the  very 
foundation  of  the  art  of  transmitting  me- 
chanical power,  and  must  prove  of  inesti- 
mable value  to  American  mill-owners,  me- 
chanics, and  operatives. 

In  the  present  volume  Mr.  Fairbairn 
gives  the  results  of  his  very  successful  prac- 
tice as  a  Millwright  and  Engineer,  during 
a  period  of  half  a  century — a  period -which 
has  contributed  more  than  any  previous  one 

to  the  manufacturing  industry  of  the  world. 

5 


6  PREFACE. 

And  as  there  is  probably  no  department  of 
practical  science  so  generally  useful,  or  per- 
haps so  little  studied,  as  the  machinery  of 
transmission,  the  American  publisher  has 
great  pleasure  in  placing  within  reach  of" 
the  intelligent  class  for  which  it  is  intended, 
in  a  compact  and  comparatively  cheap  form, 
the  author's  rich  and  valuable  experience 
on  this  important  subject. 


CONTENTS. 


CHAPTER  I. 

THE  PRINCIPLES  OF  MECHANISM. 

PAGl 

GENERAL  VIEWS,  LINK-WORK,  WRAPPING  CONNECTORS 
WHEEL-WORK,  SLIDING  CONTACT  : — 

General  Yiews  Relating  to  Machines 13 

The  parts  of  a  Machine 18 

Elementary  Forms  of  Mechanism 27 

Link-work « 22 

ELEMENTARY  FORMS  OF  MECHANISM: — 

To  construct  Watt's  parallel  motion : . . . .  31 

To  multiply  Oscillations  by  means  of  Link-work.. .  34 
To  produce  a  Telocity  which  shall  be  rapidly  re- 
tarded by  means  of  Link-work 36 

To  produce  a  reciprocating  intermittent  Motion  by 

means  of  Link-work 37 

The  Ratchet-wheel  and  Detent 39 

Intermittent  motion  produced  by  Link-work  con- 
nected with  a  Ratchet-wheel 39 

Wrapping  Connectors 40 

Speed  Pulleys 44 

Guide  Pulleys 47 

To  prevent  Wrapping  Connectors  from  Slipping. .  48 

System  of  Pulleys 51 

To  produce  a  varying  velocity  ratio  by  means  of 

Wrapping  Connectors 54 

(5) 


CONTENTS. 

run 

Wheel-w6rk  producing  Motion  by  rolling  Contact..  56 

Idle  Wheels 63 

Annular  Wheels — Concentric  Wheels 64 

Wheel-work  when  the  axes  are  not  parallel  to  each 

other 64 

Face-wheel  and  Lantern — Crown-wheels 65 

To  construct  Bevel-wheels  or  Bevel-gear  when  the 

axes  are  in  the  same. plane 66 

To  construct  Bevel-gear  when  the  axes  are  not  in 

the  same  plane 68 

Variable  motions  produced  by  Wheel-work  having 

rolling  contact 69 

Intermittent  and  reciprocating  motions  produced  by 

Wheel-work  having  rolling  contact 71 

The  A\redge  and  Movable  Inclined  Plane 74 

Sliding  Pieces  producing  motion  by  sliding  contact.  74 

The  Eccentric  Wheel 75 

Cambs,  Wipers,  and  Tappets 76 

To  find  the  curve  forming  the  groove  of  a  Camb,  so 

that  the  velocity  ratio  of  the  rod  and  axes  of  the 

Camb  may  be  constant 77 

The  Swash  Plate 80 

Construction  of  Screws 82 

The  Solid  Screw  and  Nut 85 

The  Common  Press 86 

The  Compound  Screw 88 

The  Endless  Screw 89 

The  Differential  Screw 90 

The  Archimedian  Screw  Creeper 91 

Mechanism  for  Cutting  Screws 92 

To  produce  a  changing  reciprocating  rectilinear 

motion  by  a  combination  of  the  Camb  and  Screw.  94 
To  produce  a  boring  motion  by  a  combination  of 

the  Screw  and  Toothed  Wheels: .  95 


CONTENTS.  7 

CHAPTER    II. 

ON  MACHINERY  OF  TEANSMISSION. 

PA01 

ON  WHEELS  AND  PULLIES  : — 

Wrapping  Connections 99 

Where  employed 100 

Advantages  and  Disadvantages  of 101 

Material  employed  in  the  Construction  of 101 

Strength  of 102 

Table  of  approximate  Widths  of   Leather  Straps, 
in  Inches,  necessary   to   transmit   any  required 

Number  of  Horses'  Power 103 

TOOTHED  WHEELS  : — 

Introduction  of. 104 

Construction  of  Mortise  Wheels 105 

Smeaton's  Introduction  of  Cast-iron  as  a  Material 

for  Spur  Wheels 107 

Rennie's  use  of  Cast-iron  in  all  the  details  of  Mill  work, 
as  exemplified  in  the  Construction  of  the  Albion 

Mills 107 

True  Principle  of  Construction 108 

Tooth-cutting  Machine ....    112 

SPUR  GEARING  : — 

Definitions 114 

PITCH  OF  WHEELS  : — 

Rules  for  finding  the  Pitch  and  Diameter  of  Wheels.  117 

Table  of  Constants  for  Wheel-work 118 

Rules  for  finding  the  Pitch,  Diameter,  and  Number 

of  Teeth 119 

Professor  Willis's  Method  of  graduating  the  si7*o 

of  Wheels .121 

* 

Table  showing  the  relation  of  Pitch,  Diameter,  and 
Number  of  Teeth 122,  \23 


8  CONTENTS. 

PAG* 

TEETH  OP  WHEELS: — 

The  Principles  which  determine  the  proper  Form..  124 
Formation    of    Epicycloidal     and     Hypocycloidal 

Curves 125 

Construction  of  Epicycloidal  Teeth 129 

Construction  of  Involute  Teeth 135 

Professor  Willis's  Method  of  striking  the  Teeth  of 

Wheels 140 

Odontograph 142 

General  Form  and  Proportions  of  Teeth  of  Wheels.  145 
Table  of  Proportions  of  Teeth  of  Wheels  for  aver- 
age Practice 154 

Table  giving  the  Proportions  of  the  Teeth  of  Wheels 

in  Inches  and  Thirty-seconds  of  an  Inch 156 

BEVEL  WHEELS  : — 

Examination  of  the  Curves 157 

Formation  and  Form  of  Teeth 159 

SKEW  BEVELS: — 
Definitions  and  Method  of  setting  out  the  Teeth. . .  160 

THE  WORM  AND  WHEEL  : 
Description  of  Construction 163 

STRENGTH  OP  THE  TEETH  IN  WHEELS  : — 

Rules  to  be  observed  in  Calculations 165 

Line  of  greatest  Strain 167 

Table  of  Thickness,  Breadth,  and  Pitch  of  Teeth  of 

Wheels 168 

Table  of  Relation  of  Horses'  Power  transmitted 
and  Velocity  at  the  Pitch  Circle  to  Pressure  on 

Teeth 172 

Table  showing  the  Pitch  and  Thickness  of  Teeth  to 
transmit  a  given  Number  of  Horses'  Power  at 
different  Velocities...  .  173 


CONTENTS.  9 

PAOB 

Table  showing  the  Breadth  of  Teeth  required  to 
transmit  different  Amounts  of  Force  at  a  uniform 
Pressure  of  400  Ibs.  per  inch 174 

CHAPTER  III. 

ON  THE  STRENGTH  AND  PROPORTIONS  OF  SHAFTS  : — 
The  Factory  System  necessitates  the  use  of  long 
Eanges  of  Shafts 175 

DIVISION  I. : — 

The  Material  of  which  Shafting  is  constructed  ....   177 
DIVISION  II.    TRANSVERSE  STRAIN  : — 

Resistance  to  Rupture 179 

Rules  for  the  Strength  of  Shafts 183 

Table  of  Resistance  to  Flexure.  Weights  produc- 
ing a  Deflection  of  Tz>aff  of  the  Length  in  Cast- 
iron  Cylindrical  Shafts 187 

Table  of  Resistance  to  Flexure.  Weights  produc- 
ing a  Deflection  of  yj'jjjjth  of  the  length  in 

Wrought-iron  Cylindrical  Shafts 188 

Table  of  Deflection  of  Cast-iron  Cylindrical  Shafts, 

arising  from  the  Weight  of  the  Shaft 189 

Table   of   Deflection  of  Wrought-iron  Cylindrical 

Shafts,  arising  from  the  Weight  of  the  Shaft. . . .  190 
DIVISION  III.    TORSION  : — 

Coulomb's  Deductions  and  Formula 191 

Bevan's  Values  of  Modulus  of  Torsion 192 

Wertheim's  Formulae  for  Cylindrical  Bodies 193 

R6sum6  of  Experiments  on  Cylinders  of  Circular 

Section 196 

ResumS  of  Experiments  on  the  Torsion  of  Hollow 

Cylinders  of  Copper 197 

Resum6  of  Experiments  on  the  Torsion  of  Ellipti- 
cal Bars..  .  197 


10  CONTENTS. 

MM 

Table  of  the  safe  Working  Torsion  for  Cast-iron 
Shafts 200 

Table  of  the  safe  Working  Torsion  for  Wrought- 
iron  Shafts 201 

DIVISION  IV. : — 

Velocity  of  Shafts 204 

Table  of  the  Diameter  of  Wrought-iron  Shafting  ne- 
cessary to  transmit  with  safety  various  Amounts 
of  Force 205 

DIVISION  V.     ON  JOURNALS  : — 

Length  of  Journals 207 

Ultimate  Pressure  per  Square  Inch  on  Journal ....  208 

Form  of  Journals 208 

DIVISION  VI.    FRICTION  : — 

Laws  of 209 

Kennie's  Table  of  Coefficients  of  Friction  under  Pres- 
sures increased  continually  up  to  Limits  of  Abra- 
sion   212 

DIVISION  VII.     LUBRICATION  : — 
Lubricants 213 

Method  of  effecting  complete  Lubrication 215 

CHAPTER  IV. 

ON    COUPLINGS  FOE  SHAFTS  AND  ENGAGING  AND 

DISENGAGING  GEAE. 
COUPLINGS  : — 

Primitive  Cast-iron  square  Coupling-box 216 

The  Claw  Coupling 217 

Mr.  Hewe's  Coupling 218 

The  Disc  Coupling 219 

The  Circular  Half-lap  Coupling 219 


CONTENTS.  11 

PAGg 

Bales  for  the  Proportions  of  the  Half-lap  Coupling.  220 
The  Cylindrical  Butt-end  Coupling/ 220 

DIVISION    VIII.       DISENGAGING    AND    RE-ENGAGING 

GEAR  : — 
Throwing  Wheels  out  of  Gear  with  an  Horizontal 

Lever 222 

Throwing  Wheels  out  of  Gear  with  a  Standard  or 

Pkunmerb-lock  and  Movable  Slide 223 

Disengaging  Machinery  by  the   Fast  and  Loose 

Pulley. 225 

Disengaging  Machinery  with  the  Sack  Teagle  Mo- 
tion   226 

Calendering  Marine  Friction  Clutch 227 

Friction  Cones -. 227 

Friction  Discs 228 

Friction  Couplings 229 

Disengaging  and  Re-engaging  Clutch 230 

Two  other  Forms  of  ditto 231 

Mr.  Bodmer's  Clutch 236 

DIVISION  IX.      HANGERS,  PLUMMER-BLOCKS,  ETC.,  FOE 
CARRYING  SHAFTING  : — 

Pedestal  for  supporting  Shafting  on  the  Floor 238 

Pedestal  for  bolting  Shafting  to  a  Wall 239 

Hanger  for  suspending  Shafting  from  a  Beam  in 

the  Ceiling 240 

Hanger  for  suspending  Shafting  from  the  Floor. . .  241 

Hanger  where  great  Strength  is  required 242 

Hanger  to  connect  two  or  three  Ranges  of  Shafting.  244 
Method  of  connecting  Ranges  of  Shafting  at  Right 

angles  to  each  other  by  means  of  Plummer-blocks.  246 
Table  of  the   Diameters,  Pitch,  Velocity,  etc.,  of 
„     Spur  Fly-wheels  of  the  new  Construction 250 


12  CONTENTS. 

PAGl 

MAIN  SHAFTS  : — 

Material,  Diameter,  etc. 251 

Description  of  the  Main  Vertical  Shafts . . ' 252 

Description  of  the  Method  of  Gearing  the  Saltaire 

Mills 252 

Method  adopted  to  lessen  the  Friction  on  the  Foot 

of  the  Vertical  Shaft 256 

Present  Method  by  Bevel  Wheels 258 

Transmission   of  Power  to  Machinery  at  Obtuse 

Angles  by  the  Universal  Joint 259 

Table  of  the  Length,  Diameter,  etc.,  of  Couplings, 

Coupling-Boxes,  etc 261 


PRINCIPLES  OF  MECHANISM. 


CHAPTEK    I. 

OENSRAL   VIEWS.— LINK- WORK. — WRAPPING    CONNECTORS 

WHEEL-WORK. — SLIDING    CONTACT. 

I.   GENERAL   VIEWS   RELATIVE  TO   MACHINES. 
Definitions  and  Preliminary  Expositions. 

1.  Mechanism  may  be  defined  as  the  combina- 
tion, of  parts  or  pieces   of  a  machine  whereby 
motion  is  transmitted  from  the  one  to  the  other, 

2.  When  a  body,  or  any  piece  of  mechanism, 
moves   in  a  straight  line   it  is   said  to  have  a 
rectilinear  motion,  and  when  it  moves  in  a  curved, 
line  it  is  said  to  have  a  curvilinear  motion.     When 
a  point  moves  constantly  in  the  same  path,  it  is 
said  to  have  a  continuous  motion,  but  if  it  moves 
backwards   and  forwards  it  is   said  to  have  a 
reciprocating  motion.     We  may  have  reciprocating 
rectilinear  motions  as  well  as  reciprocating  curvili- 
near motions. 

If  a  body  moves  over  equal  spaces  in  equal 
intervals  of  time,  it  has  a  uniform  motion  ;  but  if  it 
moves  over  unequal  spaces  in  equal  intervals  of 
time,  it  has  a  variable  motion, 

2  (13) 


14:  PRINCIPLES  OF   MECHANISM. 

3.  The  velocity  of  a  body  is  the  rate  at  which  it 
moves.     In  uniform  motion  the  velocity  is  con- 
stant ;   but  in  variable   motion  the  velocity  con- 
tinually  changes.     If   the   velocity   of    a  body 
increase  it  is  said  to  be  accelerated,  and  if  the 
velocity  decrease  it  is  said  to  be  retarded. 

The  motion  of  a  body  is  said  to  be  periodical 
when  it  undergoes  the  same  changes  in  the  same 
intervals  of  time. 

4.  In  order  to  express  the  velocity  of  a  body, 
we  must  have  a  certain  number  of  units  of  space 
passed   over  in   a  certain   unit   of  time.     It   is 
customary  to  take  a  foot  as  the  unit  of  space,  and 
a  second  as  the  unit  of  time. 

In  uniform  motion,  the  space  passed  over  is 
equal  to  the  product  of  the  velocity  by  the  time. 
Thus,  let  s  be  the  space  in  feet,  t  the  time  in 
seconds,  and  v  the  velocity  per  second  ;  then 


which  expresses  the  general  relation  of  space,  time, 
and  velocity,  in  uniform  motions.  Any  two  of 
these  elements  being  given  the  remaining  one 
may  be  found  ;  thus  we  have 

«  =  y...  (2),  and  «  =  £...  (3). 

5.  If  the  velocity  in  one  certain  direction  be 
taken  as  positive,  then,  that  in  the  opposite  or 
contrary  direction  will  be  negative. 


GENERAL   VIEWS.  15 

6.  If  two  wheels  perform  a  revolution  in  the 
same  time,  their  angular  velocities  are  equal,  what- 
ever may  be  the  dimensions  of  the  wheels.     The 
angular  velocity  of  a  revolving  wheel  or  rod  is  the 
velocity  of  a  point  at  a  unit  distance  from  the 
centre  of  motion.     The  wheel  or  rod  will  revolve 
uniformly  when  the  angular  velocity   is  uniform. 
If  A  be  the  angular  velocity,  r  the  radius  of  the 
wheel  or  length  of  the  rod,  v  the  velocity  at  this 
distance  from  the  centre  of  motion ;  then 

A  =  -=  (1),  and  v  =  A  r  ...  (2). 

7.  The  motion  of  wheels  is   conveniently  ex- 
pressed by  the  number  of  rotations  which  they 
perform  in  a  given  time.     Thus,  let  n  be  the  num- 
ber of  revolutions  performed  per  min.,  the  other 
notation  being  the  same  as  in  Art.  6 ;  then 

v  =  ^r*  nr  ...(1),  and  n  =  —  -  ...(2). 

oU  rt  T 

V 

Or  substituting  A  for  -.    See  formula  (I),  Art.  6, 
n  =  — (3),  and  A  =  gg  *  n  ...  (4). 

Hence  the  number  of  turns  performed  in  a 
given  time  varies  as  the  angular  velocity. 

The  number  of  turns  which  two  wheels  respec- 
tively make  in  the  same  time  is  called  their  syn- 
chronal  rotations.  Let  Q  and  q  be  the  synchronal 


16  PRINCIPLES   OF   MECHANISM. 

rotations  of  two  wheels  whose  angular  velocities 

Q  A 

are  A  and  a,  respectively ;  then  -  =  - ;  that  is, 

synchronal  rotations  are  in  the  ratio  of  the  angular 
velocities. 

Example. — Let  a  wheel  whose  radius  is  6  ft. 
perform  50  revolutions  per  min.,  required  1st,  the 
velocity  of  its  circumference,  and  2nd,  its  angular 
velocity. 

Here,  by  eq.  (1),  n  =  50,  and  r  =  6,  then 

v  =  J-  x  3-1416  x  50  x  6  =  31-416  ft.  per  sec. 

oU 

And,  by  eq.  (4),  A  =  -^  x  31416  x  50  =  5-236. 

8.  If  v  and  v  be  the  velocities  of  two  parts  of  a 

v 

piece  of  mechanism,  then  -  is  the  velocity  ratio  of 

these  parts.  Let  s  and  s  be  the  corresponding 
spaces  described  in  the  same  time,  then  when  the 
motion  is  uniform 

V         s 

-  =  —  =  a  constant, 

v         s 

that  is,  when  the  velocities  are  uniform,  the  velo- 
city ratio  is  constant. 

9.  If  the  velocity  ratio  of  the  two  parts  remains 
constant,   then   however   variable   the   velocities 

v        s 

themselves  may  be,  we  still  shall  have  -  =  - 

v         s 

where  s  and  s  are  the  entire  spaces  described  i 
the  same  interval  of  time. 


GENERAL   VIEWS.  17 

10.  When  a  body  moves  with  a  variable  motion, 
its  velocity  at  any  instant  is  determined  by  the 
rate  at  which   it   is   moving   at  that   particular 
instant,  that  is,  by  the  space  which  it  would  move 
over  in  one  second,  supposing  the  motion  which 
it  then  has  to  remain  constant  for  that  time. 
Variable  motions   may  be  graphically  repre- 
Fig.  i.  sented,  by  taking  the 

abscissa  of  a  curve 
equal  to  the  units  of 
time,  and  the  ordinates 
equal  to  the  units  of 
the  corresponding 
velocities.  Thus  let  A 
B  be  equal  to  the  units  of  velocity  at  the  com- 
mencement of  the  motion ;  A  c  the  units  in  inter- 
val of  time,  C  D  the  units  in  the  corresponding 
velocity ;  and  so  on ;  then  the  area  of  the  curved 
space  A  B  D  F  E  will  be  equal  to  the  space  described 
in  the  interval  of  time  represented  by  A  E. 

If  the  motion  be  uniform  the  curve  B  D  F  will 
become  a  straight  line  parallel  to  o  x,  and  the 
space  described  in  any  given  time  will  be  repre- 
sented by  the  area  of  a  rectangle,  whose  length  is 
equal  to  the  units  of  time,  and  breadth  equal  to 
the  units  of  velocity. 

If  the  motion  be  uniformly  accelerated  or  re- 
tarded, the  curve  B  D  F  will  become  a  straight  line 
inclined  to  the  axis  ox,  and  the  space  described 
in  this  case,  will  be  represented  by  the  area  of  a 
2* 


18  PRINCIPLES   OF   MECHANISM. 

trapezoid,  whose  base  is  equal  to  the  units  of  time 
and  parallel  sides  respectively  to  the  velocity  at 
the  commencement  and  end  of  that  time. 

11.  THE  PARTS  OP  A  MACHINE.  —  A  machine 
conists  of  three  important  parts. 

(1.)  The  parts  which  receive  the  work  of  the 
moving  power  —  these  may  be  called  RECEIVERS 
of  work. 

(2.)  The  parts  which  perform  the  work  to  be 
done  by  the  machine  —  these  may  be  called  WORK- 
ING PARTS,  or  more  simply,  OPERATORS. 

(3.)  The  mechanism  which  transmits  the  work 
from  the  receivers  to  the  working  parts  or  opera- 
tors —  these  pieces  of  mechanism  may  be  called 

COMMUNICATORS   OF  WORK,   Or  the   TRANSMISSIVE 


The  form  of  the  mechanism  must  always  be 
determined  from  the  relation  subsisting  between 
the  motions  of  the  receivers  and  operators. 

If  there  were  no  loss  of  work  in  transmission 
(from  friction,  etc.)  the  work  applied  to  the  re- 
oeiver  would  always  be  equal  to  the  work  done 
by  the  operator.  Thus,  let  P  be  the  Ibs.  pressure 
applied  to  the  receiver,,  and  s  the  space  in  feet 
which  it  moves  over  in  a  certain  time  ;  P  the  Ibs. 
pressure  produced  at  the  working  part,  and  s  the 
space  in  feet  which  it  moves  over  in  the  sam 
time;  then,  neglecting  the  loss  of  work  by  friction 
we  have  — 


GENERAL   VIEWS.  19 

Work  applied  to  the  receiver  =  work  done 
upon  the  operator, 

or  PXS  =  PiXS1...(l). 

However,  it  must  be  borne  in  mind,  that  the 
actual  or  useful  work  done  by  a  machine  is  always 
a  certain  fractional  part  of  the  work  applied ;  this 
fraction,  determined  for  any  particular  machine,  is 
called  the  modulus  of  that  machine.  If  m  be  put 
for  this  modulus,  then  we  have  from  eq.  (1) 

raxpxs  =  P1xs1  ...(2). 

In  treating  of  the  motion  of  these  parts  of  a 
machine  it  is  generally  most  convenient  to  find  an 
expression  for  their  proportional  velocities.  Thus, 
let  V  be  the  velocity  of  the  receiver,  and  yl  that 

V 
of  the  operator ;   then  —  is  their  velocity  ratio. 

See  Art.  8. 

It  must  be  observed,  that  this  velocity  ratio  is 
not  at  all  effected  by  the  actual  velocities  of  the 
parts,  provided  the  velocity  ratio  of  the  mechanism 
be  constant  for  all  positions.  In  the  more  ordi- 
nary pieces  of  mechanism  (such  as  common 
toothed  wheels,  wheels  moved  by  straps,  levers, 
etc.)  the  velocity  ratio  is  constant,  that  is  to  say, 
it  remains  the  same  for  all  positions  of  the 
mechanism. 

In  eq.  (1)  s  may  be  taken  as  the  velocity  of  the 
power  P,  estimated  in  the  direction  in  which  it 


20  PRINCIPLES   OF   MECHANISM. 

acts,  and  8t  that  of  the  resistance  P^  then  this 
equality  becomes — 

px  v  =  PiX  V!...(3), 

p  v 

or,  —  -  =  — •  =  the  velocity  ratio  ...  (4). 

P 

Now  —  is  called  the  advantage  gained  by  the 

machine,  or  the  number  of  times  that  the  resist- 
ance moved  is  greater  than  the.  power  applied. 
Hence  the  advantage  gained  by  a  machine,  irre- 
spective of  friction,  etc.,  is  equal  to  the  velocity 
of  the  power  divided  by  the  velocity  of  the  resist- 
ance, or  the  velocity  ratio  of  the  power  and  resist- 
ance. 

This  is  called  the  principle  of  virtual  velocities. 
Workmen  express  this  dynamic  law  by  saying, 
*'  What  is  gained  in  power  is  lost  in  speed." 

12.  The  DIRECTIONAL  RELATION  of  the  motion 
of  the  receiver  and  the  operator  admits  of  every 
possible  variation.  It  may  be  constant  or  it  may 
.be  variable.  By  the  intervention  of  mechanism 
rectilinear  motion  may  be  converted  into  curvili- 
near motion,  and  conversely ;  reciprocating  recti- 
linear or  circular  motion,  may  be  converted  into 
continuous  circular  motion,  and  conversely;  and 
so  on  to  the  various  possible  combinations  of 
which  the  cases  admit.  These  directional  changes 
are  so  important,  in  a  practical  point  of  view,  that 
some  eminent  writers  on  mechanism  have  made 


GENERAL   VIEWS.  21 

them  the  basis  of  the  classification  of  mechanism. 
But,  however  eligible  in  a  practical  point  of  light 
such  a  classification  may  be,  there  is  complexity 
in  its  application,  which  renders  it  less  suitable 
for  scientific  purposes  than  that  method  of  classifi- 
cation which  is  based  upon  the  nature  or  mode  of 
action  of  certain  elementary  pieces  of  mechanism 
which  enter,  more  or  less,  into  every  mechanical 
combination. 

Elementary  Forms  of  Mechanism. 

13.  In  analysing  the  parts  of  a  machine  we  find 
motion  transmitted  by  jointed  rods  or  links,  by 
straps  and  cords,  by  wheels  rolling  on  other  wheels, 
and  by  pieces  of  various  forms  sliding  or  slipping 
on  other  pieces.     Hence  we  have  the  following 
elementary  forms  of  mechanism : 

(1.)  Transmission  of  motion  by  jointed  rods, — 
LINK-WORK. 

(2.)  By  straps,  cords,  etc., — WRAPPING  CON- 
NECTORS. 

(3.)  By  wheels  or  curved  surfaces,  revolving  on 
centres,  rolling  on  each  other, — WHEEL-WORK. 

(4.)  By  pieces  of  various  forms,  sliding  or  slip- 
ping on  each  other, — SLIDING-PIECES. 

14.  The  velocity  ratio,  as  well  as  the  directional 
relation,  in  an  elementary  piece  of  mechanism  may 
be  either  constant  or  varying.     The  number  of 
combinations   of  which  these   elementary  pieces 
admit,  is  almost  unlimited.     The  eccentric  wheel 


22  PRINCIPLES   OF   MECHANISM. 

is  a  combination  of  sliding  pieces  and  link-work. 
The  common  crane  is  a  combination  of  wheel- 
work,  link- work,  and  wrapping  connectors ;  and 
so  on  to  other  cases. 

A  train  of  mechanism  must  be  supported  by 
some  frame  work  ;  the  train  of  pieces  being  such, 
that  when  the  receiver  is  moved  the  other  pieces 
are  constrained  to  move  in  the  manner  determined 
by  the  mode  of  their  connection.  Eevolving 
pieces,  such  as  wheels  and  pulleys,  are  so  connect- 
ed with  the  frame  that  every  portion  of  them  is 
constrained  to  move  in  a  circle  round  the  axis ; 
and  sliding  pieces  are  constrained  to  move  in 
straight  lines  by  guides. 

Mechanism  is  to  a  great  extent  a  geometrical 
inquiry.  The  motion  of  one  piece  in  a  train  may 
differ,  both  in  kind  and  direction,  from  the  motion 
of  the  next  piece  in  the  series :  these  changes  are 
effected  by  the  geometrical  construction  of  the 
pieces,  as  well  as  by  their  mode  of  connection. 
The  investigation  of  the  law  of  these  changes  con- 
stitutes one  of  the  chief  objects  of  the  principles 
of  mechanism. 


II.   ON  LINK-WORK. 


15.  If  a  bent  rod  or  lever  A  c  B  turn  upon  the 
centre  c,  the  velocities  of  the  extremities  A  and  B 


LINK  WOKK. 


23 


will  be  to  each  other  in  the  ratio  of  their  distances 

from  the  centre  of  motion  c,  that  is, 

velocity  A circum.  cir.  A  Qt A  c 

velocity  B      circum.  cir.  B  Q        B  c 


Fig  2. 


It  is  not  necessary  that  the  arms  A  c  and  B  c 
should  be  in  the  same  plane.  Thus  let  c  D  be  an 
axis  round  which  the  arms  A  E  and  B  F  revolve, 
then, 

velocity  A      perpend,  dist.  A  from  the  axis 
velocity  B     perpend,  dist.  B  from  the  axis 

Tig.  4. 


16.  Let  A  B,  B  D,  D  E,  be  a  series  of  levers  turn- 
ing on  the  fixed  centres  c,  Q,  and  E  ;  then  when 
the  arcs,  through  which  the  extremities  A  and  B 
are  moved,  are  small  the  velocity  ratio  will  be  ex 
pressed  by  the  following  equality : — 


24:  PRINCIPLES   OF   MECHANISM. 

Velo.  A        A  C.      B  Q.      D  R 
Velo.  E        B  C.      D  Q.      E  R 

that  is  to  say,  the  velocity  ratio  of  P  and  Pl  is  found 
by  taking  the  product  of  the  lengths  of  the  arms  ly- 
ing toward  P,  and  dividing  by  the  product  of  those 
lying  toward  PI. 

17.  To  find  the  velocity  ratio  of  the  rods  A  B  and 
C  D,  turning  on  the  fixed  centres,  A  and  D ;  and  con- 
nected by  the  link  B  C. 

Fig.  5. 


Through  the  centres  A  and  D,  draw  the  straight 
line  D  E  A,  cutting  C  B  in  E ;  and  from  A  and  D  let 
fall  the  perpendiculars  A  G  and  D  K  upon  C  B,  or 
it  may  be  upon  c  B  produced.  Then 

ang.  velo.  DC     A  G 

=  — ...  (1); 

ang.  velo.  A  B     D  K 


LINK  WORK.  25 

that  is  to  say,  the  angular  velocities  of  the  rods  D  c 
and  A  B  are  to  each  other  in  the  inverse  ratio  of  the 
perpendiculars    let  fall  from   their  respective   axes 
upon  the  direction  of  the  link. 
Similarly  we  also  have : 

ang.  velo.  DC A  E 

ang.  velo.  A  B      D  E     ^  ' ' 

t  iat  is  to  say,  the  angular  velocities  of  the  rods  D  0 
and  A  B  are  to  each  other  in  the  inverse  ratio  of  the 
segments  into  which  the  link  divides  the  line  joining 
their  axes. 

These  velocity  ratios  are  obviously  varying, 
depending  upon  the  relative  positions  of  the  rods. 

18.  THE  CRANK  AND  GREAT  BEAM. — Let  A  B 
represent  one  half  of  the  great  beam  of  a  steam- 
engine,  D  c  the  crank,  and  B  c  the  connecting  rod. 
Putting  ft  for  the  angle  D  c  B,  and  ft  for  the  angle 
ABC;  then 

velo.  crank sin  j3t 

velo.  beam       sin  ft   * V  V 

When  th,e  connecting  rod  B  c  is  very  long  as 
compared  with  the  length  of  the  crank  D  c,  then  ft 
is  nearly  constant,  being  nearly  equal  to  90°,  in 
this  case,  eq.  (I)  becomes 

velo.  crank  _      1 
velo.  beam  ~~  sin  ft 

The  crank  must  be  in  the  same  straight  line 
with  the  connecting  rod,  at  the  highest  and  lowest 
3 


26  PRINCIPLES  OF   MECHANISM. 

points  of  the  stroke  of  the  beam,  and  then  0  =  0. 
In  these  positions  the  crank  is  said  to  be  at  its 
dead  points. 

The  velocity  ratio,  expressed  by  eq.  (2),  will  be 
a  maximum  when  /3  =  0,  that  is,  the  velocity  of  the 
crank  will  be  a  maximum  when  it  is  in  its  dead 
points.  "When  0  =  90°,  or  when  the  crank  is  at 
right  angles  to  the  connecting  rod,  then  the 
velocity  of  the  crank  is  a  minimum. 

If  B  =  A  B,  or  one-half  the  length  of  the  great 
beam ;  r  =  1)  c,  the  length  of  the  crank ;  and  A  = 
the  angular  oscillation  of  the  beam,  or  the  whole 
angle  described  by  the  beam  in  one  stroke ;  then 

A 

r  =  B  sin  £  ...  (3) 

which  expresses  the  length  of  the  crank  in  terms 
of  the  radius  of  the  beam  and  angle  of  its  stroke. 

A  double  oscillation  of  the  beam  produces  one 
complete  rotation  of  the  crank,  or  conversely,  taking 
the  crank  as  the  driver,  each  rotation  of  the  crank 
produces  a  double -oscillation  in  the  beam. 

From  eq.  (1)  it  follows,  that  the  velocity  of  the 
crank  is  equal  to  the  velocity  of  the  beam,  when 
|3  =  ^  or  angle  D  c  B  is  equal  to  angle  ABC;  that 
is,  when  the  position  of  the  crank  is  parallel  to 
that  of  the  beam.  j 

By  this  form  of  the  crank  the  reciprocating  circu- 
lar motion  of  the  extremity  of  the  beam  is  changed 
into  a  continuous  circular  motion ;  and  conversely 


LINK  WORK. 


27 


a   continuous   circular  motion   is    changed   into  a 
reciprocating  circular  motion. 

19.  To  determine  the  various  relations  of  position 
and  velocity  of  the  CRANK  and  PISTON  in  a  locomo- 
tive engine.  Fig.  e. 

Here  the  connecting  rod, 
D  E,  is  attached  to  the  ex- 
«.  tremity  of  the  piston  rod, 
p  D,  and  the  length  of  the 
stroke  of  the  piston  is  equal 
to  double '  the  length  of  the 
crank,  F  E.  Moreover,  the 
centre,  F,  of  the  crank  is  in 
the  same  straight  line  with 
the  axis  of  the  cylinder  or 
the  direction  of  the  piston 
rod. 

Let  Z=D  E,  tlie  length 
of  the  connecting  rod ; 

^  =  p  D,  the  length  of 
the  piston  rod ; 

r  =  F  E,  the  length  of 
the  crank ; 

k  =  F  D,  the  varying  dis- 
tance of  the  extremity  of 
the  piston  rod  from  the 
axis  of  the  crank ; 

h  =  the  corresponding 
height  of  the  stroke  of  the  piston ; 

0  =  the  varying  angle,  FED,  which  the  crank 
forms  with  the  direction  of  the  connecting  rod. 


28  PRINCIPLES   OF   MECHANISM. 

(1.)  The  velocity  ratio  of  the  crank  and  piston  is 
expressed  by  the  following  equality  : 

velo.  crank  k 

~  =  ^—.  -  —(1),  or 
velo.  piston       I  sin  e 

~"sin  0 

where  0  in  eq.  (2)  is  put  for  angle  E  F  D  ;  that  is, 
the  angle  which  the  crank  makes  with  the  direc- 
tion of  the  piston  rod. 

This  latter  form  of  the  expression  is  the  same 
as  that  given  in  eq.  (2),  Art.  18. 

(2.)  When  the  piston  is  at  the  bottom  point  of 
its  stroke,  its  distance  from  F  =  FE-fED-f-Dp 
=  r  +  I  +  ?,  ;  also  FD  =  FE  +  DE  =  r-fZ. 

When  the  piston  is  at  the  middle  point  of  its 
stroke,  then  F  D  =  E  D  ;  that  is  to  say,  in  this 
position  of  the  piston  D  E  F  will  be  an  isosceles 
triangle. 

(3.)  Tlie  position  of  the  crank  at  any  point  of  the 
stroke  of  the  piston  is  determined  by  the  two  following 
general  equations  :  — 


003  '  = 


When  the  piston  is  at  the  middle  point  of  its 
stroke,  then  h  =  r,  and  eq.  (4)  becomes 


cos  e  =  ~  -(5). 


LINK  WORK. 


29 


When  the   crank   is   at   right    angles    to  the 
connecting  rod,  0  =  90°,  and  then  we  find  from 


Fig.  7. 


_ 
h  =  r  +  l—  Vi*  +  l  .-.(6). 

This  expression  is,  obviously,  less  than  r,  or 
half  the  whole  stroke  of  the  piston.  Hence  it 
appears  that  the  crank  is  at  right  angles  with  the 
connecting  rod,  before  the  piston  has  attained  the 
middle  point  of  its  upward  stroke. 

20.  Fig.  7  shows  how  a  rotation  of  the  axis  A  is 
transmitted  to  another  c,  by  means  of  the  two 
equal  cranks  A  B  and  c  D,  connected  by  the  con- 
necting rod  D  B,  whose  length  is 

equal  to  the  distance  A  c,  between 
the  two  axes.  In  all  positions 
of  the  cranks,  the  figure  A  B  c  D 
will  be  a  parallelogram,  and  the 
velocity  of  D  will  always  be  equal 
to  the  velocity  of  B,  and  the  mo- 
tion of  the  axis  c  will  be  exactly 
the  same  as  that  of  the  axis  A. 

21.  Two  sets  of  cranks  may  be 
placed  upon  the  axes,  having  the 
cranks  on  each  axis  at  right  an- 
gles to  each  other,  similar  to  the 
mode  of  connecting  the  wheels  of 

a  locomotive  engine,  as   shown  in  fig.  8,  where 
the  cranks  are  formed  by  bending,  or  loops  made 
in  the  axes.     These  axes  must  be  parallel  to  each 
3* 


30 


PRINCIPLES   OF    MECHANISM. 


other,  and  the  connecting  rods  must  also  be  of 
equal  lengths. 

The  advantage  of  this  combination  consists  in 
maintaining  a  constant  moving  pressure,  by  which 
means  an  equable  motion  is  sustained  without  the 
aid  of  the  inertias  of  the  machinery. 


Fig.  9. 


22.  The  double  universal  joint,  represented  in 
fig.  9,  furnishes  another  example  of  link-work, 
for  transmitting  motion  from  one  axis  to  another 
axis.  This  useful  piece  of  mechanism  should  be 
constructed,  so  that  the  extreme  axes,  A  B  and  c  D, 
would  meet  in  a  point,  if  produced,  and  the  angles 
which  they  respectively  make  with  the  central 
line  of  the  intermediate  piece,  E  F  H  G,  shall  be 
equal  to  each  other. 


LINK  WORK. 
Fig.  10. 


31 


TO  CONSTRUCT  WATT'S  PARALLEL  MOTION. 

23.  This  beautiful  and  useful  piece  of  mechan- 
ism is  formed  by  a  combination  of  link- work. 

L*t  A  B.and  c  D  (see  figs.  10  and  11)  be  two 
rods,  turning  on  the  fixed  centres  A  and  D,  and 
connected  together  by  «&•  "• 

the  short  link  c  B; 
then  when  motion  is 
given  to  the  rods, 
there  is  -a  certain 
point,  E,  in  the  link 
c  B,  which  will  move, 
or  very  nearly  move,  in  a  straight  line.  In  mat- 
ter of  fact  the  path,  or  locus,  of  this  point  is  a 
curve  of  the  fourth  degree ;  but  when  the  motion 
of  the  rods  is  limited,  and  their  lengths  are  con- 
siderable, as  compared  with  the  length  of  their 
connecting  link,  this  path  becomes  almost  exactly 
a  straight  line. 


32  PRINCIPLES   OF   MECHANISM. 

In  fig.  11,  C  B  K  R  is  a  parallel  frame  of  links; 
to  the  joint  R  is  attached  the  piston  rod  R  P  of  the 
steam  engine ;  and  to  the  point  E  is  attached  the 
piston  rod  of  the  air-pump. 

(1.)  To  find  the.  point  E  (see  fig.  10)  to  which  the 
air-pump  rod  must  be  attached,  having  given  t/ie 
radius  rod  C  D,  the  link  c  B  or  Q  G,  and  the  rod  A  B 
or  A  G  forming  a  part  of  the  great  beam. 

Let  D  Q,  A  G  be  an  extreme  position  of  the  rods. 
Let  the  rods  be  moved  to  the  position  A  B  c  D, 
where  the  link  c  B  is  perpendicular  to  A  B  and 
D  c.  Produce  B  c,  meeting  the  link  Q  G  in  the 
point  E  ;  then  E  will  be  that  point  of  the  link  which 
will  most  nearly  move  in  a  vertical  straight  line. 
The  ratio  of  Q  E  to  G  E  is  generally  expressed  by 
the  following  equality : — 

/  r  sin  H  \ 


GE         r     \      .   A 


where  R  =  AB,  r  =  DC,  a=  angle  C  D  "Q,  and  A  = 
angle  BAG. 

Practically  the  link  Q  G  or  c  B  deviates  very 
little  from  the  vertical;  and  the  angles  a  and  A  are 

a  .    A 

small  ;  hence,  r  sin  ^  =  R  sin  ^  very  nearly  ;    in 

this  case,  therefore,  eq.  (1)  simply  becomes 


GE 


LINK  WORK.  33 

and  from  this  equality  we  readily  find, 
DQXGQ 

G  E  =  —  -  -"(3), 

D  Q  +  A  Q        V   * 

•which  gives  the  position  of  the  point  E,  as  re- 
quired. 

f1  O 

When  D  Q  =  A  G,  then  G  E  =  -^-  ,  that  is  to  say, 

JL 

in  this  case,  the  point  E  is  at  the  middle  of  the 
link  Q  G  or  c  D. 

Example,  —  Let  A  B  or  A  G  =  5  ft.  ;  D  c  or  D  Q  — 
4  ft.;  and  C  B  or  G  Q  =  l-5  ft.;  then  by  eq.  (3)  we 
have, 

_  4x1-5          2 
:  ~     =  3  ft 


(2.)  -To  find  the  length  of  the  radiiJLS  rod  D  c  (see 
fig.  13),  when  the  divisions  A  B  and  B  K,  on  the  learn 
are  given. 

In  this  case, 

A  Ba 

The  radius  rod,  D  c  =  -   -  •••  (4). 
B  K 

When  A  B  =  B  K  ;  then  D  c  =  A  B  ;  that  is,  in 
this  case,  the  radius  rod  will  be  equal  to  the  divi- 
sion A  B  on  the  beam. 

Example.  —  Let  A  B  =  6  ft.,  and  B  K  =  4  ft.;  then 
by  eq.  (4)  we  have, 

6a 

The  radius  rod,  D  c  =  -7-  =  9  ft. 
4 


PRINCIPLES   OF   MECHANISM. 


To  multiply  Oscillations  ly  means  of  Link-work. 

24.  Fig.  12  represents  a  system  of  links  B  A  c, 
c  D,  and  D  E,  turning  on  the  fixed  centres  A  and  E, 
and  having  the  arms  A  B  and  A  c  united  to  the 
same  centre  A.  The  construction  is  such,  that 

while  the  rod  A  B 
makes  a  single  os- 
cillation from  B 
to  I,  the  rod  E  D 


oscillation,  viz., 
from  D  to  F,  and 
back  from  F  to  D. 
The  oscillations 
of  A  B  are  pro- 
,  duced  by  the  ro- 
tation of  a  crank 
(see  Art.  17),  or 
by  any  other 
means. 

The  conditions 
of  the  construc- 
tion may  be  stat- 
ed as  follows : 

Given  the 
lengths  of  the 

arms  A  c  and  E  D,  the  lengths  or  angles  of  their 
oscillations,  and  the  length  of  the  connecting  link 
C  D,  to  construct  the  mechanism,  so  that  the  rod 


LINK  WORK.  85 

E  D  shall  perform  two  oscillations  whilst  A  B 
makes  one. 

Let  B  A  c  be  the  position  of  the  bent  lever  at 
the  commencement  of  the  upward  oscillation. 
Draw  A  I  and  A  H,  making  the  angles  B  A  I  and  0 
A  H  each  equal  to  the  angle  of  the  oscillation. 
From  A  as  a  centre,  with  A  B  and  A  c  as  radii,  de- 
scribe the  arcs  B  I  and  C  H.  Through  A  draw  A 
G  F  bisecting  the  angle  c  A  H  cutting  the  arc  c  H 
in  G.  On  A  G  F  take  A  F  equal  to  the  sum  of  the 
rods  A  c  and  c  D,  and  make  F  D  equal  to  the  given 
length  of  the  oscillation  of  E  D.  From  D  and  F 
as  centres,  with  a  radius  equal  to  the  length  of  the 
rod  E  D,  describe  circles,  cutting  each  other  in  E  ; 
then  E  will  be  the  centre  of  the  rod  E  D,  which 
will  perform  two  oscillations,  whilst  the  rod  A  B 
makes  one. 

When  A  B  and  A  c  are  in  the  middle  points  of 
their  oscillations,  the  rod  E  D  will  have  the  posi- 
tion E  F,  that  is,  it  will  have  performed  a  complete 
upward  oscillation.  When  A  B  and  A  c  have  per- 
formed the  remaining  halves  of  their  oscillations, 
the  rod  E  F  will  have  returned  to  the  original  po- 
sition, that  is,  it  will  have  performed  a  complete 
downward  oscillation. 

In  like  manner  the  oscillations  may  be  further 
multiplied,  by  connecting  E  D  with  another  series 
of  links. 


36  PRINCIPLES   OP   MECHANISM. 

To  produce  a  Velocity  which  shall  be  rapidly 

retarded,  by  means  of  Link-work. 
25.  In  fig.  13,  R  A  c  and  E  D  represent  two  rods, 
turning  on  fixed  centres  A  and  E,  and  connected 
by  a  link  c  D ;  the  rod  E  D  is  supposed  to  oscil- 
late uniformly  between  the  positions  E  D  and  E  F. 
Now  the  construction  is  such  as  to  produce  a 
rapidly  retarded  motion  of  the  rod  R  c  in  moving 

Tig.  13. 


from  the  position  R  A  c  to  the  position  SAB,  and 
conversely. 

The  conditions  of  the  construction  may  be 
stated  as  follows : 

Given  the  rods  E  D  and  D  c  in  position  and  mag- 


LINK  WORK.  87 

nitude,  the  angle  of  oscillation  DBF,  and  the 
length  of  the  rod  A  0,  to  construct  the  mechanism. 

Bisect  the  arc  D  F  in  G,  and  then  bisect  the  arc 
F  G  in  K ;  through  the  points  K  and  B,  draw  the 
straight  line  K  E  c ;  from  D  and  K  as  centres,  with 
a  radius  equal  to  the  length  of  the  link  D  c,  de- 
scribe arcs,  cutting  K  E  c  in  the  points  C  and  B  ; 
from  B  and  c  as  centres,  with  a  radius  equal  to  the 
length  of  the  rod  A  c,  describe  arcs  cutting  each 
other  in  the  point  A ;  then  A  will  be  the  centre  of 
the  rod  AC. 

When  the  rod  E  D  arives  at  the  position  E  G,  the 
rod  BAG  will  have  the  position  SAB  very  nearly, 
and  it  will  have  moved  with  a  rapidly  retarded 
motion.  During  the  remaining  half  of  the  oscil- 
lation G  F,  the  rod  SAB  will  remain,  virtually, 
stationary. 

This  piece  of  mechanism  was  first  employed  by 
Watt  for  opening  the  valves  of  the  steam  engine. 

To  produce  a  Reciprocating  Intermittent  Motion  by 
means  of  Link-work. 

26.  A  B  and  c  D  (fig.  14.)  are  two  rods,  turning 
on  the  fixed  centres  A  and  D,  and  connected  by  a 
link  B  C.  The  rod  A  B  is  made  to  oscillate  between 
the  positions  A  B  and  A  I,  by  means  of  a  crank 
and  connecting  rod.  The  construction  of  the 
mechanism  is  such,  that  the  rod  D  c  will  oscillate 
between  the  positions  D  c  and  D  F,  but  with  an 
intermittent  motion. 
4 


38  PRINCIPLES   OF    MECHANISM. 

The  conditions  of  the  construction  may  be 
stated  as  follows  : 

Given  the  rods  A  B,  B  c,  and  c  D  in  position  and 
magnitude,  to  construct  the  mechanism. 


.  14. 


From  A  as  a  centre,  with  the  radius  A  B,  de- 
scribe the  arc  B  I ;  through  c  and  A  draw  the 
straight  line  c  A  G,  meeting  the  arc  in  G  ;  make 
G  E  equal  to  one  third  the  arc  G  B,  and  on  the  arc 
take  G  I  equal  to  G  E ;  on  the  line  G  A  c  take  G  F 
equal  to  B  c ;  then  half  the  chord  B  I  will  give  the 
length  of  the  crank,  and  c  F  will  be  the  arc 
through  which  the  rod  D  c  oscillates. 

Bisecting  the  angle  B  A  E,  &c.,  the  position  of 
the  rod  D'  c'  is  found,  which  being  connected  with 
B,  by  the  link  B  c',  will  oscillate  exactly  in  a  con- 


LINK  WORK.  39 

trary  manner  to  that  of  the  rod  D  c,  that  is  to  say, 
when  D  c  is  stationary  D'  c'  will  be  in  motion,  and 
conversely. 

When  the  point  B  arrives  at  E,  the  rod  D  c  will 
have  completed,  practically,  its  oscillation,  and 
there  it  will  remain  stationary  until  the  rod,  turn- 
ing on  the  centre  A,  returns  from  the  position  A  I 
to  A  E. 

The  Ratchet-wheel  and  Detent. 

27.  In  fig.  15,  A  represents  the  ratchet-wheel, 
and  D  the  detent,  falling  into  the  angular  teeth  of 
the  ratchet,  thereby  admitting  the  Fig-  * 
wheel  to  revolve  in  the  direction 

of  the  arrow,  but  at  the  same  time 
preventing  it  from  revolving  in 
the  opposite  direction. 

In  certain  kinds  of  machinery, 
the  action  of  the  moving  force  un- 
dergoes periodic  intermissions ;  in 
such  cases  the  ratchet  and  detent  are  used  to  pre- 
vent the  recoil  of  the  wheels,  and  sometimes  to 
give  an  intermittent  motion  to  the  wheel,  as  in 
the  following  example. 

Intermittent*Mbtion  produced  l>y  Link-work  connected 
with  a  Ratchet-wheel. 

28.  B  E  is  a  rod,  turning  on  the  fixed  centre  B, 
to  which  a  reciprocating  motion  is  given  by  the 


40  PRINCIPLES   OF   MECHANISM. 

connecting  rod  c  of  a  crank,  or  by  any  other 
means ;  E  F  is  a  click,  jointed  to  the  rod  B  E  at  its 
Fig.  16.  extremity,  and  gives  mo- 

tion to  the  ratchet-wheel 
A.  At  each  upward  stroke 
of  the  rod  B  E,  the  click 
E  F,  acting  upon  the  saw- 
like  teeth  of  the  ratchet- 
wheel,  causes  it  to  move 

round  one  or  more  teeth ,  and  when  the  extremity 
F  of  the  click  is  drawn  back  by  the  descent  of  the 
lever  B  E,  it  will  slide  over  the  bevelled  sides  of 
the  teeth  without  giving  any  motion  to  the  wheel, 
so  that  at  every  upward  stroke  of  the  rod  c  the 
ratchet-wheel  will  be  moved  round  and  it  will  re- 
main at  rest  during  every  downward  stroke  of  the 
rod.  Thus  the  reciprocating  motion  of  the  con- 
necting rod,  c,  will  produce  an  intermittent  circular 
motion  in  ttys  axis  A. 


III.   ON  WRAPPING   CONNECTORS. 

29.  When  the  moving  force  of  the  machinery  is 
not  very  great,  cords,  belts,  and  other  wrapping 
connectors,  are  most  usually  employed  in  transmit- 
ting motion  from  one  revolving  axis  to  another. 

30.  The  endless  cord  or  belt  A  B  c  D,  represented 
in  figs.  17  and  18,  passes  round  the  wheels,  A  B 


WRAPPING   CONNECTORS. 


41 


and  c  D,  revolving  on  the  parallel  axes  R  K  and 
Q  F,  and  transmits  motion  from  the  axis  Q  F  to  the 
axis  R  K,  with  a  constant  velocity  ratio.  In  all 
such  cases  the  motion  is  entirely  maintained  by  the 
frictional  adhesion  of  the  cord  or  belt  to  the  sur- 
face of  the  wheel. 


Fig.  17. 


Fig.  18. 


When  the  cord  passing  round  the  wheels  is 
direct,  as  in  fig.  17,  the  motions  of  the  wheels  take 
place  in  the  same  direction,  and  when  the  cords 
cross  each  other,  as  in  fig.  18,  the  motions  of  the 
wheels  take  place  in  opposite  directions. 

If  the  wheel  c  D  makes  one  revolution,  then. 


No.  revo.  A  B  = 


circum.  c  D       radius  c 


circum.  A  B       radius  A  B 


(1). 


Or  putting  R  and  r  for  the  radii  of  the  wheels 
4* 


42  PRINCIPLES   OF   MECHANISM. 

C  D  and  A  B  respectively,  and  Q  and  q  for  their  re- 
spective synchronal  rotations,  then 

2  =  1  ...  (2). 

Q         R         V   J 

t 

Example. — If  the  radius  of  the  wheel  C  D  be  12 
inches,  and  that  of  A  B  9  inches,  what  will  be  the 
least  number  of  entire  revolutions  which  they 
must  make  in  the  same  time  ? 

Here,  by  eq.  (2),  we  have 

q  __  R  __  12  _  4 

Q  ~~  r  =  ~~  9"  =  ~  3' 

12  4 

The  fraction  -^  reduced  to  its  least  terms,  is  g, 

therefore  the  least  number  of  synchronal  rotations 
are  4  and  3,  that  is  to  say,  whilst  the  wheel  C  D 
makes  3  rotations,  the  wheel  A  B  will  make  4. 

31.  Fig.  19  represents  a  system  of  three  revolv- 
ing axes,  in  which  motion  is  transmitted  from  one  to 
Fig.  19.  the  other,  by  means 

of  a  series  of  belts. 
The  belt  being 
direct  in  the  wheels 
A  and  D  c,  t  h  e  i  r 
axes  will  move  in 
the  same  direction,  but  as  the  belt  crosses  in  pass 
ing  from  D  c  to  H  G,  their  axes  will  move  in  oppo 
site  directions. 


WRAPPING   CONNECTORS.  43 

Here,  whilst  the  axis  B  makes  one  rotation,  the 

rad.  H  G  x  rad.  t>  c 
No.  rotations  A  =  ^^^^  ' 

Or  putting  R!  =  rad  D  c,  R,  =  rad.  H  G,  &c., 
TI  =  rad.  I  K,  ra  =  rad.  E  F,  &c.,  and  putting  q  and  Q 
for  the  synchronal  rotations  of  the  first  and  last 
axes  respectively;  then 

q  R1XR,XR,X&C. 

—    ——    ~ n *  "    (  &  )• 

Q          rt  x  ra  x  ra  x  &c. 

Example. — In  the  mechanism  represented  in 
fig.  19,  let  R!  =  8,  Rg  =  15,  TI  =  5,  rt  =  4 ;  required 
the  least  number  of  entire  rotations  performed  in 
the  same  time  by  the  axes  A  and  B. 

Here,  by  eq.  (2)  we  have, 

q  __  8  X  15  _  6 
Q       5x4  "~  1 

that  is,  whilst  the  axis  B  makes  one  revolution, 
the  axis  A  will  make  six. 

32.  In  raising  buckets  from  deep  wells  or  from 
pits,  a  continuous  cord  coils  round  an  axle  or  a 
drum  wheel,  as  the  case  may  be,  the  full  bucket 
being  attached  to  one  end  of  the  cord  and  the 
empty  bucket  to  the  other  end ;  the  rotation  of  the 
axle  coils  up  the  cord  to  which  the  full  bucket  is 
attached  and  at  the  same  time  uncoils  the  cord  to 
which  the  empty  one  is  attached,  so  that  whilst 
the  former  is  ascending  the  latter  is  descending. 


44 


PRINCIPLES   OF   MECHANISM. 


Fig.  20. 


Speed  Pulleys. 

33.  Fig.  20  represents  an  arrangement  of  speed 
pulleys ;  A  B  and  c  D  are  two  parallel  axes  upon 
each  of  which  is  fixed  a  series  of  pulleys,  or 
wheels,  adapted  for  a  belt  of 
given  length,  so  that  it  may  be 
shifted  from  one  pair  of  wheels  to 
any  other  pair,  say  for  example, 
from  the  pair  a  a^  to  the  pair  c  Ci. 
In  order  to  suit  this  arrangement, 
if  the  belt  be  crossed,  the  sum  of 
the  diameters  of  any  pair  of  pulleys 
must  be  a  constant  quantity,  that  is 
to  say,  it  must  be  equal  to  the  sum 
of  the  diameters  of  any  other  pair. 
By  this  contrivance,  a  change  in 
the  velocity  ratio  of  the  two  axes 
is  produced  by  simply  shifting  the 
belt  from  one  pair  to  another. 

In  practice  it  is  customary  to  make  the  two 
groups  of  pulleys  exactly  alike,  the  smallest  pulley 
of  one  being  placed  opposite  to  the  largest  of  the 
other. 

In  a  group  of  speed  pulleys,  let  s  =  the  constant 
sum  of  the  diameters  of  the  driver  and  follower, 
D  =  the  diameter  of  the  follower,  and  Q  q  the 
number  of  their  synchronal  rotations  respectively. 

*u      Q       d       i 
then  -  =  -,  and 

q        D 


WRAPPING   CONNECTORS.  45 


^       QXS  •      i 

a  =  -  —  »  or  more  simply, 


=  s  —  D  ...(2). 

Example.  —  Kequired  the  diameters  of  a  pair  ot 
speed  pulleys,  when  the  sum  of  the  diameters  is 
30  inches,  and  the  driver  makes  two  revolutions, 
whilst  the  follower  makes  three. 

Here  s  =  30,  Q  =  2,  and  q  =  3  ;  then  by  eq.  (1) 
and  (2)  we  have 

3  x  30 

D=  —  F—  =18  in.;  and  d  =  30  —  18  =  12  in. 
5 

If  the  constant  sum  of  the  diameters  of  a  group 
of  5  pairs  of  speed  pulleys  be  12  inches,  and  the 
diameters  of  the  pulleys  a1?  bu  Ci,  ck,  *\,  be  10,  8, 
6,  4,  and  2  inches  respectively,  then  the  diameters 
of  the  pulleys  a,  b,  c,  d,  e,  will  be  2,  4,  6,  8,  and 
10  inches  respectively  ;  and  as  the  strap  is  shifted 
from  one  pair  of  wheels  to  another,  the  relative 
velocities  of  the  axes  CD  and  AB  will  be  as  the 
numbers  §,  |,  1,  2,  and  5. 

34.  It  is  customary  to  construct  the  pairs  of 
speed  pulleys  so  that  the  rotations  of  the  follower 
may  be  increased  or  decreased  in  a  certain  geo- 
metric ratio.  Thus,  if  r  be  this  ratio,  then  for  5 
pairs  of  speed  pulleys  we  shall  have  the  series  of 

terms      '  -'  1,  r,  r1,  for  the  different  values  of  -, 


46  PRINCIPLES   OF   MECHANISM. 

tha  ratio  of  the  synchronal  rotations  of  each  pair. 
Or,  generally,  if  n  be  the  number  of  pairs,  then 

_,  _.  n — 3    n — 1 

~2~    ~~2~ 

_?— 1,    j^±,  — ,  r  ,   r    ,    will     be    the    different 

values  of  —     In  this  case,  let  D1;  D2,  ... ,  D,,  =  the 

diameters  of  the  1st,  2d,  ... ,  and  nth  pulleys,  re- 
spectively, on  the  driving  axis ;  and  these  symbols, 
taken  in  a  reverse  order,  will  be  the  correspond- 
ing diameters  of  the  pulleys  on  the  driven  axis ; 

s                        s 
then  D!  =  -        —7     D*= — -,    and  so  on: 

n — 1 ,  n — .) 

1-fr  2  1+r  2 

moreover  we  have  D  =  s — Dy  DW_X  =  s  —  D2,  and 
so  on. 
Example. — To  find  the  diameters  of  a  set  of  5 

Q 

pairs  of  speed  pulleys,  so  that  values  of.-    (the 

ratio  of  the  synchronal  rotations  of  the  different 
pairs)  shall  have  the  common  ratio  of  f,  the  con- 
stant sum  of  the  diameters  of  each  pair  being  26 
inches. 

Here  r  =  f ,  n  =  5,  and  S  =  26,  then  from  the 
foregoing  formula?  we  find, 

26  26 

-I  Q  .     ^     .   1  ^3  • 

J.O  ,     U2  112      10o  J 


26 
D» =   i  i /2\o  =  13  ;  and  so  on. 


WRAPPING   CONNECTORS.  47 

But  the  remaining  diameters  will  be  better 
found  as  follows : 

D6  =  26— 18  =  8;  D4=26  — 151  =  10?,. 

85.  Two  plain  cones,  having  their  axes  parallel, 
as  shown  in  fig.  21,  will,  obviously  answer  the 
same  purpose  as  the  ordi-  Fig  21 

nary  form  of  speed  pulleys. 
The  slant  faces  of  the  cones 
may  be  formed  by  any  con- 
tinuous curve ;  but  with  this 
condition — that  the  sum  of 
the  diameters  at  every  po- 
sition of  the  band  shall  be 
a  constant. 

Guide  Pulkys. 

36.  By  the  intervention  of  guide  pulleys  the  di- 
rection of  cords  may  be  changed  into  any  other 
direction.     Thus,  by  means  of 

the  guide  pulleys  B  and  c,  the 
motion  of  the  cord  in  the  di- 
rection c  D  is  changed  into  the 
direction  A  B. 

The  cords  D  C  and  C  B 
should  be  in  the  plane  of  the 
pulley  C;  and  the  cords  C  B 
and  B  A  should  be  in  the  plane  of  the  pulley  B. 

37.  Two  guide  pulleys,  E  and  H,  may  be  em- 
ployed to  transmit  motion  from  the  wheel  A  to 


PRINCIPLES   OF   MECHANISM. 


rig.  23. 


the  wheel  B,  when  the  axes  of  these  wheels  have 
any  given  direction. 

Let  E  H  be  the  line 
where  the  planes, 
passing  through  the 
two  wheels,  intersect 
each  other.  In  this 
line  assume  any  two 
convenient  points  E 
and  H ;  in  the  plane 
of  the  wheel  A  draw 
the  tangents  E  C  and 
H  D ;  and  in  the  plane 
of  the  wheel  B  draw 
the  tangents  E  F  and 
H  G ;  then  C  E  F  G  H  D 

will  be  the  path  of  the  endless  cord,  which  will  be 
kept  in  this  path  by  a  guide  pulley  at  E,  in  the 
plane  of  C  E  F,  and  another  guide  pulley  at  H,  in 
the  plane  of  D  H  G. 

The  relative  velocities  of  the  axes  A  and  B  de- 
pend entirely  upon  the  ratio  of  the  radii,  A  D  and 
B  G,  of  the  two  wheels.  See  Art.  30. 


To  prevent  Wrapping  Connectors  from  Slipping. 

38.  The  slip  of  the  band  on  the  wheel,  when  it 
is  not  excessive,  is  in  many  cases  rather  an  advan- 
tage than  otherwise;  but  when  motion  is  to  be 
transmitted  from  one  wheel  to  another  according 


WRAPPING    CONNECTORS.  49 

to  some  given  exact  ratio,  gearing  chains  of  various 
forms  are  employed  as  the  wrapping  connectors. 

39.  In  some  cases  the  links  of  the  gearing  chain 
lay  hold  of  pins  or  teeth  formed  upon  the  wheel, 
as  shown  in  fig.  25.  In  other  cases,  the  links  of 
t!ie  gearing  are  joined  together,  something  like  a 

Fig.  24.  Fig.  25. 


watch  chain,  and  carry  teeth  which  pass  into  cer- 
tain notches  made  at  corresponding  distances  on 
the  edge  of  the  wheel,  as  shown  in  fig.  24. 


Fig.  26.  Fig.  27. 


40.  When  a  belt  moves  a  conical  wheel,  it  always 
happens  that  the  belt  gradually  moves  toward  the 
broad  end  of  the  wheel;  this  is  owing  to  the  belt 
5 


50 


PRINCIPLES   OF   MECHANISM. 


Fifr.  28. 


being  more  stretched  on  that  side  than  it  is  on 
the  other. 

41.  This  property  enables  us  to  construct  a 
wheel  so  that  a  belt  shall  not  shift  on  its  edge  ; 
this  is  simply  effected   by  making   the   edge  to 
swell  a  little  in  the  middle,  as  shown  in  fig.  27. 

42.  When  two  rollers   have  to  make  only  a 
limited  number  of  revolutions  in  each  direction, 

the  slip  of  the  cord  may 
be  prevented  by  having 
a  cord  coiled  round  each 
end  of  the  rollers  in  op- 
posite directions,  so  that 
while  one  cord  is  coiled 
on  one  extremity  of  the 

roller,  the  other  cord  is  uncoiled  from  the  other 

extremity,  as  shown  in  fig.  28. 

Pig.  29. 


43.  By  a  similar  arrangement  of  cords  on  the 
cylinder  E  F  (see  fig.  29),  a  reciprocating  motion 


WRAPPING   CONNECTORS. 


51 


of  this  cylinder  will  produce  a  back  and  forward 
motion  of  the  carriage  A  B. 

Systems  of  Pulleys. 

44.  A  system  of  pulleys  must  at  least  contain 
one  movable  pulley.     When  a  wheel,  forming  a 
part  of  a  system  of  wheels  connected  together  by 
cords,    has   a   progressive   motion,    it    materially 
affects  the  velocity  ratio  of  the  receiver  and  the 
operator  of  the   mechanism.     There  are  a  great 
many  different  systems  of  pulleys,  but  they  all  de- 
pend upon  the  different  combinations  of  movable 
and  fixed  pulleys,  and  the  different  modes  of  redu- 
plication of  a  cord. 

45.  In  this  system  of  pulleys 
there  is  one  movable  block  and  a 
single  continuous  cord  with  three 
duplications,   so   that   whilst  the 
moving  force  P  acts  by  one  cord, 
the  movable  block  with  its  load 
is  suspended  by  six  cords ;  if  w 
ascend  one  foot,  each  of  these  cords 
will   be  shortened   one  foot,  and 
therefore  the  cord  P  will  be  length- 
ened six  feet ;  that  is  to  say,  the 
velocity  of  P  will  be  six  times  that  of  w. 

46.  In  the  system  of  pulleys  represented  in  fig. 
31,  there  are  two  distinct  cords  and  two  movable 
pulleys,  A  and  B,  making  two  duplications  of  cord ; 


Fig;.  30. 


52 


PRINCIPLES   OF   MECHANISM. 


then  if  A  ascend  one  foot,  B  must  ascend  two  feet, 
and  the  cord  at  P  must  be  lengthened  four  feet; 
that  is,  the  velocity  of  P  will  be  four  times  the 
velocity  of  w. 

Generally,  if  there  are  n  moveable  pulleys  in. 
Buch  a  system,  then, 

velo.  P  =  2n  x  velo.  w. 


V\g.  32. 


Tig.  33. 


47.  The  system  of  pulleys  represented  in  fig.  32, 
contains  two  movable  pulleys,  one  fixed  pulley, 
and  two  single  cords.     In  this  case  the  velocity 
ratio  of  P  to  W  is  as  four  to  one. 

48.  Fig.  33  represents  a  similar  system  of  pul 


WRAPPING   CONNECTORS.  53 

leys,  in  which  the  velocity  ratio  of  P  to  w  is  as 
five  to  one. 

In  all  these  systems  of  pulleys  the  velocity  ratios 
are  constant. 

49.  In  the  compound  wheel  and  axle,  repre- 
sented in  fig.  34:,  the  axle  is  made  of  different 
thicknesses,  as  at  A  and  Fig.  34. 

B,  and  a  continuous  cord 
coils  round  these  parts 
in  different  directions, 
and  passes  round  the 
wheel  of  the  movable 
pulley  D.  In  one  revo- 
lution of  the  wheel  c  P, 
the  space  moved  over  by  the  pulley  D  is  equal  to 
half  the  difference  of  the  circumferences  of  the 
axles  A  and  B.  Putting  Ri  for  the  radius  of  the 
wheel  c  P,  R  for  the  radius  of  the  axle  A,  and  r  for 
the  radius  of  the  axle  B ;  then  we  have  for  the 
velocity  ratio 

velo.  P         2  R! 
velo.  w  ~  ~  R  —  r 

If  Rl=10,  R  =  4,  r  =  3f ;  then 

velo.  P         2x10 
velo.  w  ~~  4 —  3f 

This  piece  of  mechanism  belongs  to  a  class  which 
produces  what  has  been  called  differential  motions, 
5* 


54:  PRINCIPLES   OF    MECHANISM. 

their  object  being  to  produce  a  slow  and  definite 
motion  in  a  body  by  the  most  simple  and  practi- 
cable means. 

TO   PRODUCE   A  VARYING   VELOCITY  RATIO  BY 
MEANS  OF  WRAPPING   CONNECTORS.  , 

50.  To  find  the  ratio  of  the  angular  velocities 
of  two  eccentric  wheels,  moved  by  a  cord  wrap- 
ping over  each.  ' 
Let  D  c  be  a  cord  wrapping  round  the  wheels, 
Fig.  35.                  whose  axes  of  motion  are 
A  and  B ;  their  line  C  D  will 
be  a  tangent   to  the  two 
curves  forming  the  edges 
of  the   wheels.     On   D  c 
produced  let  fall  the  per- 
pendiculars A  Q  and  B  K; 
then   the   velocity   of  the 
cord,  in  this  position  of  the 
wheels,  will  be  equal  to  the 
velocity  of  the  point  Q,  and  at  the  same  time  it  will 
also  be  equal  to  the  velocity  of  the  point  K :  hence 

we  find, 

angular  velocity  AC         B  K 


angular  velocity  B  D        A  Q 


(i); 


that  is  to  say,  the  angular  velocities  are  inversely  as 
the  perpendiculars  let  fall  upon  the  cord  from  the 
of  motion. 


WRAPPING   CONNECTORS. 


55 


51.  Let  B  be  a  movable  pulley  suspended  from 
the  continuous  cord  P  A  B  c,  Fi*-  36- 

passing  over  a  fixed  pulley 
A,  and  attached  to  a  point 
c  in  the  same  horizontal 
line  with  A.  Let  fall  B  D 
perpendicular  to  A  c  ;  then 
B  c  will  always  be  equal  to 
B  A,  and  B  will  move  in  a  vertical  line  B  D.  Hence 
we  find, 


velocity  P 


B  D 

=  2  x  — 


velocity  w 

This  expression  may  be  put  in  the  following 
trigonometrical  form : 

velocity  P 

-  =  2  x  cos  P  A  B  ...  (2). 
velocity  w 

52.  Fig.  37  represents  a  simple  and  ingenious 
contrivance  for  communicating  a  varying  velocity 
to  the  axis  B,  by  means  Fig.  37. 

of  an  endless  band  Q  K  C, 
passing  over  an  eccen- 
tric wheel  A,  a  pulley 
B  K,  and  a  stretching 
pulky  c.  The  curve  of 
the  eccentric  wheel  A, 
must  be  such  as  to  pro- 
duce the  varying  velo- 
city required.  The 
weight  w,  attached  to  the  stretching  pulley  C,  keeps 


56 


PRINCIPLES   OF  MECHANISM. 


the  band  constantly  stretched,  so  that  whatever 
may  be  the  velocity  of  the  cord  upon  leaving  the 
eccentric  wheel,  it  communicates  the  same  velocity 
to  the  circumference  of  the  pulley  B  K.  From  the 
axis  A  let  fall  A  Q  perpendicular  to  the  cord  Q  K ; 
then  by  eq.  (1),  Art.  50,  the  velocity  ratio  may 
be  expressed  as  follows : 

ang.  velo.  axis  A        B  K 
ang.  velo.  axis  B  ~  ~  A  Q 

Let  the  axis  A  revolve  uniformly,  and  let  the 
radius  B  K  of  the  pulley  be  given ;  then 

The  ang.  velo.  axis  B  will  vary  as  the  perpend.  A  Q. 


IV.  ON  WHEEL-WORK  PRODUCING  MOTION  BY 
ROLLING  CONTACT  WHEN  THE  AXES  OF  MOTION 
ARE  PARALLEL. 

53.  Two  wheels,  E  and  F,  in  contact  with  each 
Kg.  38.  other,  revolve  on  the  par- 

allel axes  A  B  and  c  D ; 
now  if  the  wheels  are  in 
contact  in  any  one  posi- 
tion, they  will  also  be  in 
contact  in  every  other  po- 
sition, and  their  circum- 
ferences will  roll  upon 
each  other,  so  that  if  the 
driver  F  revolve  on  its  axis  c  D,  it  will  communi- 


WHEEL   WORK. 


57 


Fitr.  39. 


cate  a  rotatory  motion  to  the  follower  E  in  a  con- 
trary direction,  by  the  frictional  adhesion  of  the 
parts  successively  brought  in  contact.  The  edges 
of  these  wheels  must  have  the  same  velocity,  and 
therefore  their  angular  velocities  will  be  inversely 
as  their  radii. 

54.  In  order  to  render  the  transfer  of  motion 
perfectly  exact,  the  edges  of  the  wheels  are  formed 
into  teeth,  placed  at  equal 
distances  from  each  other, 
so  that  when  one  wheel  is 
turned,  its  teeth  success- 
ively enter  into  the  spaces 
formed  on  the  edge  of  the 
other  wheel.  Thus,  even 
with  slight  errors  of  con- 
struction, one  wheel  can 
not  escape  from  the  other, 
which  may  happen  in  the 
case  of  simple  rollers. 

The  numbers  of  teeth  in  the  wheels,  acting  upon 
each  in  this  manner,  are  in  proportion  to  their 
radii.  Thus,  let  the  radius  of  the  wheel  A  be  15 
inches,  that  of  B  6  inches,  and  let  B  contain  8 
teeth;  then 

15 
No.  teeth  in  A  =  8  x  -~-  =  20. 


Or,  generally,  if  R  and  r  be  put  for  the  radii  of  the 


58  PRINCIPLES   OF   MECHANISM. 

wheels,  and  N  and  n  the  number  of  their  teeth 
respectively;  then 


Hence  angular  velocities,  as  well  as  the  synchronal 
rotations,  of  wheels,  may  be  expressed  in  terms  of 
their  numbers  of  teeth  ;  thus  we  have  — 

ang.  velo.  A  _n 

ang.  velo.  B       x'"  ^  '' 

synchronal  rotation  A         Q        n 

also,  —  :  --  ,  or  -  =  -  ...  (3). 

synchronal  rotation  B         q       N 

Example.  —  Required  the  least  number  of  teeth 
in  the  wheels  A  and  B,  so  that  B  shall  make  105 
revolutions  per  min.  and  A  only  40. 

,Q,  n         40         8 
Here  by  eq.  (3),  -  =  —  —  =  —  -  ; 
1  N        105       21  ' 

that  is,  B  will  contain  8  teeth  and  A  21  teeth. 

The  form  which  must  be  given  to  the  teeth 
of  wheels,  so  as  to  maintain  a  perfect  rolling 
contact,  will  be  explained  in  another  part  of  this 
work. 

55.  If  the  wheel  A  be  the  driver  then  B  will 
be  called  the  follower.     Wheels   acting  in   this 
manner  are  sometimes  called  spur-wheels.     Small 
toothed  wheels  are  called  pinions  ;  thus  B  may  be 
called  a  pinion  in  relation  to  A. 

56.  .  Toothed  wheels  are  said  to  be  in  gear  when 


WHEEL   WOBK.  59 

their  teeth  are  engaged  together,  and  they  are  said 
to  be  out  of  gear  when  they  are  separated. 

57.  In  the  train  of  wheels  represented  in  fig.  40, 

Kg.  40. 


let  Nx,  N2,  N,,  &c.,  be  the  number  of  teeth  in  the 
driving  wheels,  and  Wi,  nt,  n3,  &c.,  the  number  in 
the  driven  wheels  ;  Qi  =  the  no.  of  rotations  of  the 
first  axis,  Q  =  the  no.  of  the  second  axis,  and  so 
on,  performed  in  the  same  time  ;  then 

Qm  +  1        ^.N^y.....*, 
-  —  -  '  .  ..(1). 

Qx  7h.W2.W3.  ...  Um 

This  equality  may  be  expressed  in  language  as 
follows  :  —  The  ratio  of  the  synchronal  rotations  of  the 
last  and  first  axes,  is  equal  to  tlie  continued  product 
of  the  number  of  teeth  in  the  driving  wheels  divided 
by  the  continued  product  of  the  number  of  teeth  in  the 
driven  wheels. 

Similarly  we  have, 


v 
x 


Qi         Qi      Q3 

which  may  be  expressed  in  language  as  follows  :  — 
The  ratio  of  the  synchronal  rotation  of  the  first  and 


60  PRINCIPLES   OP   MECHANISM. 

last  axes,  is  equal  to  the  product  of  the  separate  syn- 
chronal  ratios  of  the  successive  pairs  of  axes. 

The  number  of  axes  in  this  combination  is 
always  one  more  than  the  number  of  pairs  of 
wheels. 

It  is  evident,  from  eq.  (1),  that  the  drivers  and 
followers  may  be  placed  in  any  order  in  a  train 
of  wheel-work  without  changing  the  velocity 
ratios  of  the  first  and  last  axes. 

Example. — Let  the  number  of  pairs  of  drivers 
and  followers  be  3,  that  is,  let  m  =  3,  NI  =  16, 
N,  =  15,  N,  =  14,  HI  =  7,  n-i  =  6,  ria  =  5 ;  required 
the  least  number  of  synchronal  rotations  of  the 
first  and  last  axes  in  the  train  of  wheels. 

Here  by  eq.  (1)  we  have — 

04  _  16  x  15  x  14  _  16^ 

oi ":  T  x  6  x  5     :T; 

that  is,  whilst  the  first  axis  makes  one  revolution, 
the  last  will  make  sixteen. 

58.  If  the  number  of  teeth  in  a  driving  wheel 
be  some  exact  multiple  of  the  number  of  teeth  in 
the  follower,  then  the  same  teeth  will  come  into 
contact  in  every  revolution  of  the  driver.  Thus 
if  the  driver  contains  30  teeth  and  the  follower  6, 
then  the  same  teeth  will  come  into  contact  at 
every  revolution  of  the  driver.  This  arrangement 
of  teeth  is  preferred  by  the  clock  and  watchmaker ; 
but  the  millwright  would  add  one  tooth,  called  the 
HUNTING  COG,  to  the  large  wheel,  that  is,  he 


WHEEL  WORK.  61 

would  have  31  teeth  in  the  driver  and  6  in  the 
follower,  because  31  and  6,  being  prime  to  each 
other,  and  at  the  same  time  nearly  in  the  same 
ratio  as  30  and  6,  the  same  pair  of  teeth  would  not 
come  again  into  contact  until  the  large  wheel  had 
made  6  revolutions,  and  the  small  one  31. 

59.  Eq.  (3),  Art.  53,  enables  us  readily  to  find 
the  number  of  revolutions  which  the  wheels  must 
make  in  order  that  the  same  teeth  may  come  again 
into  contact  with  each  other ;  for  it  is  only  neces- 

M 

sarv  to   reduce  the  fraction  -  to  its  least  terms, 

N 

and  the  denominator  of  this  reduced  fraction  will 
give  the  number  of  revolutions  of  the  driving 
wheel  as  required.  Thus,  let  N  =  144,  and  n  =  54, 

then  -=:-—==-;  that  is,  the  driver  must  make  3 
q      14      o 

complete  revolutions,  or  the  follower  8,  before  the 
same  teeth  can  again  come  into  contact. 

60.  In  a  combination  of  wheels,  whose  motions 

are  expressed  by  the  equality  -  =  — — -*.    an    in- 

J  Qx      fk.n,' 

definite  number  of  values  may  be  assigned  to  the 
numbers  of  teeth,  which  shall  produce  a  given  syn- 
chronal  ratio  of  the  first  and  last  axes ;  but  if  wx 
and  nt  be  given,  and  N!  and  N2  be  comprised  within 
certain  given  limits;  then  a  limited  number  of 
values  may  be  found  for  N!  and  N,. 

Thus,  for  example,  let  -3  =  60,  wx  =  Wj  =  8,  and 

Qi 
6 


62  PRINCIPLES   OF   MECHANISM. 

the  values  of  NI  and  Na  not  to  exceed  100  nor  to 
be  less  than  40. 
Here  we  have — 

Nl'N'-60- 
8^T8~ 

/.  N!  .  N,  =  60  x  64 ; 

hence,  Nt  may  be  60  and  Ns  may  be  64 ;  but  in 
order  to  determine  all  the  combinations,  we  must 
put  the  product,  60  x  64,  into  prime  factors,  and 
then  distribute  these  factors  into  different  groups 
answering  to  the  limiting  values  of  Na  and  N3. 
Here,  60  x  64  =  2'  X  3  X  5  ;  hence  we  have 

1st  combination,  (24  X  3)  x  (2*  x  5)  =  48  x  80 ; 

2d  combination,  (25  x  3)  x  (2*  x  5)  =  96  x  40; 

3d  combination,  2'  X  (21  X  3  x  5)  =  64  x  60. 
61.  When   all   the  drivers  contain  the   same 
number  of  teeth,  and  also  the  followers,  then  eq. 
(1),  Art.  57,  becomes 


Qi 

By  means  of  this  formula  we  may  readily  de- 
termine the  least  number  of  axes  requisite  for 
producing  a  given  synchronal  ratio  of  rotation 
between  the  first  and  last  axes,  when  the  number 
of  teeth  in  the  drivers  cannot  exceed  NH  and  the 
number  in  the  followers  cannot  be  less  than  n\. 

Find  m,  in  eq.  (1),  equal  to  the  highest  whole 
number,  which  does  not  make  the  right  member 
greater  than  the  left ;  then  the  least  number  of 


WHEEL   WORK.  63 

axes  will  be  m  +  2.  But  if  wr,  a  whole  number, 
can  be  found  so  as  to  make  the  right-hand 
member  exactly  equal  to  the  left,  then  in  this  case, 
the  least  number  of  axes  will  be  m  +  1. 

Example. — Kequired  the  least  number  of  axes 
in  a  train  of  wheels  which  shall  cause  the  last  axis 
to  revolve  180  times  as  fast  as  the  first  axis, 
allowing  that  none  of  the  drivers  can  contain  more 
than  54  teeth,  and  none  of  the  followers  less 
than  9. 

Here,  we  must  finct  the  greatest  whole  number 

(54  \m 
—  I  or  (6)m  shall  not  exceed  180. 

This  value  of  m  is  obviously  2  ;  and  the  least 
number  of  axes  will  be  4. 


Idle  Wheels. 

62.  The  wheel  c  placed  between  two  other 
wheels,  A  and  B,  does  not  affect 

Tig.  41. 

the    velocity    ratio     of    these 

wheels ;  and  hence  the  wheel  C 

is  called  an  idle  wheel.     This 

intermediate    wheel,    however, 

causes  the  wheels  A  and  B  to 

revolve   in  the   same  direction, 

whereas  if  A  and  B  were  in  contact  they  would 

revolve  in  opposite  directions. 


PRINCIPLES   OF   MECHANISM. 


Fig.  42. 


Fig.  43. 


Annular  Wheels. 

63.  Fig.  42  represents  an  annular  wheel  A,  hav- 

ing its  teeth  cut  on  the  internal 
edge  of  the  annul  us  or  rim.  The 
toothed  wheel  B,  revolving  with- 
in the  annular  wheel  A,  causes 
it  to  revolve  in  the  same  direc- 
tion; whereas  two  ordinary  spur 
wheels  revolve  in  opposite  direc- 
tions. 

• 

Concentric  Wheels. 

64.  When  two  separate  wheels  revolve  about 

the  same  centre  of  motion, 
they  are  called  concentric 
wheels.  The  pinion  D  is 
fixed  to  the  axis  F  E,  whilst 
the  concentric  wheel  c  is 
fixed  to  a  tube  or  cannon,  N, 
which  revolves  freely  upon 
the  axis  F  E.  The  driving 
wheels,  A  and  B,  fixed  to  the 
parallel  axis  H  G,  communicate  the  relative  velo- 
cities to  the  axis  F  E  and  to  the  cannon  N. 

Wheel  work  when  the  axes  are  not  parallel  to  each 
other. 

65.  When  the  axes  of  two  wheels  are  not  paral- 
lel to  each  other,  motion  is  generally  communi- 
cated from  the  one  to  the  other  by  bevel  wheels  or 


WHEEL   WORK. 


65 


Fig.  44. 


bevel  gear.  When  the  axes  are  perpendicular  to 
each  other,  the  face  wheel  and  lantern,  and  the  crown 
wheel  are  frequently  employed. 

Face  Wheel  and  Lantern. 

66.  In  fig.  44,  F  represents  a  face  wheel,  with  its 
lantern  L.  Here 
motion  is  trans- 
mitted from  the 
vertical  axis  A  B 
to  the  horizontal 
axis  AC.  The 
teeth  F  on  the  face 
of  the  face  wheel 
are  called  cogs, 
which  are  usually 

made  of  iron,  whilst  the  round  staves  forming  the 
teeth  of  the  lantern,  L,  are  made  of  hard  wood. 
The  axes  A  B  and  c  D  should,  when  produced,  in- 
tersect in  a  point. 


Grown  Wheels. 

67.  Fig.  45  represents 

a  crown  wheel  B,  with  its 

pinion   A,   having    their 

axes  at  right  angles  to 

each   other.     The  teeth 

of  the  crown  wheel  are 

cut  on  the  edge  of  a  hoop, 

the  plane  of  which  is  at 

6* 


Fig.  45. 


66  PRINCIPLES   OF   MECHANISM. 

right  angles  to  its  axis,  and  the  pinion  is  thicker 
than  wheels  are  commonly  made. 

CASE  I.  To  construct  Bevel  Wheels  or  Bevel  Gear 

when  the  axes  are  in  the  same  plane. 
68.  Let  A  c  and  A  B  be  two  axes  of  rotation,  in, 
the  same  plane,  and  cutting' 

Fif .  46. 

each  other  in  the  point  A. 
On  these   axes   two   right 
cones,  A  D  F  and  A  D  E,  may 
be  formed,  touching  each 
other   in   the   line   A  H  D ; 
and  also  two  right  frusta, 
D  F  G  H  and  D  H  K  E,  of  these 
cones  may  be  formed. 
Now,  if  the  frustum  D  F  G  H  revolve  on  its  axis 
B  A,  it  will  communicate,  by  rolling   contact,  a 
rotatory  motion  to  the  frustum  D  H  K  E  upon  its 
axis  c  A. 

These  frusta  of  cones  will  obviously  perform 
their  rotations  in  the  same  time  as  the  ordinary 
spur  wheels  previously  described. 

On  the  surfaces  of  these  frusta  a  series  of  equi- 
distant teeth  are  cut,  directed  to  the  apex  A  of  the 
cones,  so  that  a  straight  line  passing  through  the 
apex  to  the  outline  of  the  teeth  upon  the  bases  D  F 
and  D  E  of  the  frusta  shall  touch  the  teeth  in  every 
part  as  shown  in  the  diagram. 

Wheels  cut  in  this  manner  are  called  bevel  gear. 
Two  wheels  of  this  construction  will  always 


WHEEL   WORK. 


67 


Fig.  47. 


transfer  motion,  with  a  constant  velocity  ratio, 
from  one  axis  to  the  other,  provided  these  axes 
meet  each  otner  in  ,a  point,  which  point  being 
always  made  the  apex  of  the  frusta  forming  the 
bevel  of  the  wheels. 

69.  General  problem. — Given  the  radii  of  two 
bevel  wheels,  and  the  position  of  their  axes,  to 
construct  the  frusta  forming  the  wheels,  the  two 
axes  being  in  the  same  plane. 

Let  A  B  and  A  c  be  the  position  of  the  axes  cut- 
ting each  other  in  A. 
Draw  I J  parallel  to 
A  B  at  a  distance  equal 
to  the  radius  of  the 
wheel  on  the  axis  A  B ; 
and  draw  M  L  parallel 
to  A  c,  at  a  distance 
equal  to  the  radius  of 
the  wheel  on  the  axis 
A  c,  cutting  the  line  I J 
in  the  point  D.  From  the  point  D,  draw  DBF  per- 
pendicular to  A  B,  and  DOB  perpendicular  to  A  c. 
Take  B  F  equal  to  B  D,  and  c  E  equal  to  c  D.  Join 
A  E,  AD,  and  A  F.  At  a  distance  equal  to  the  thick- 
ness of  the  wheel,  draw  H  G  parallel  to  D  F,  cutting 
A  D  in  H ;  and  through  H,  draw  H  K  parallel  to  D  E. 
Then  D  F  G  H  and  D  H  K  E  will  be  the  frusta  re- 
quired. 


68 


PRINCIPLES   OF   MECHANISM. 


CASE  IT.  To  construct  Bevel  Gear  when  the  axes  are 
not  in  the  same  plane. 

70.  This  is  usually  done  by  introducing  an  in- 
termediate wheel  with  two  frusta  formed  upon  it, 
one  frustum  rolling  in  contact  with  the  driving 
wheel,  and  the  other  frustum  in  contact  with  the 
driven  wheel. 

71.  Let  A  B  and  c  D  be  the  direction  of  the  given 
axes ;  take  A  D  as  a  third  axis,  meeting  the  axes 

A  B  and  c  D  at  any 
convenient  points,  A 
and  D;  then  A  will 
be  the  vertex  of 
two  rolling  frusta 
of  cones  G  and  H, 
and  D  will  be  the 
vertex  of  two  other 
rolling  frusta  of 
cones  I  and  K. 
Whilst  the  intermediate  axis,  with  its  two  frusta 
of  cones,  revolves,  the  teeth  of  the  frustum  H  will 
have  a  rolling  contact  with  the  teeth  of  the  frus- 
tum G,  and  at  the  same  time  the  teeth  of  frustum 
I  will  have  a  rolling  contact  with  the  teeth  of  the 
frustum  K ;  and  thus  motion  will  be  transmitted 
from  the  axis  A  B  to  the  axis  C  D  with  a  constant 
velocity  ratio.  , 

Let  Q!  and  Q3  be  the  number  of  rotations  per- 
formed by  the  axes  A  B  and  c  D  respectively  in  the 


WHEEL   WORK.  69 

same  time ;  N\  =  the  number  of  teeth  in  the  bevel 
wheel  G  ;  ftt  =  the  number  in  the  edge  H ;  N2  = 
the  number  in  the  edge  I ;  and  n^  =  the  number 
in  the  bevel  wheel  K  ;  then, 

Q,==N1.N2 

Q!      n,  .  V        '' 

which  is  similar  to  the  expression  given  in  eq.  (1), 
Art.  57.  "When  nt  =  N2,  then  this  equality  be- 
comes, 

?•  =*'...  (2) 
Qi      n, 

In  this  case  the  intermediate  bevel  wheel,  I  H, 
may  be  regarded  as  an  idle  wheel. 

VARIABLE  MOTIONS  PRODUCED  BY  WHEEL  WORK 
HAVING  ROLLING  CONTACT. 

72.  Two  curved  wheels,  E  p  and  F  p,  having 
rolling  contact,  revolve  on  the  axes  A  and  B.  In 
order  that  these  wheels  may 

Fig  49 

roll  on  each  other  without 
slipping,  or  without  produc- 
ing any  strain  upon  the  axes 
A  and  B,  these  axes  must  al- 
ways be  in  the  line  of  contact 
A  p  B,  and  if  the  curve  p  E  on 
the  one  wheel  be  equal  to  the 
curve  P  F  on  the  other  wheel, 
the  sum  of  the  lines  A  E  and 
B  F  must  always  be  equal  to  A  B,  the  distance  be- 


* 
70  PRINCIPLES   OF   MECHANISM. 

tween  the  centres  of  motion.  Yarious  curves  may 
be  constructed,  having  this  property.  For  exam- 
ple, two  equal  ellipses,  E  P  and  F  p,  revolving  on 
their  foci,  A  and  B,  and  having  A  E  and  B  F  in  the 
line  of  their  major  axes,  will  have  a  perfect 
rolling  contact.  Two  equal  logarithmic  spirals 
have  also  the  same  property. 

Let  D  p  c  be  the  common  tangent  to  the  point 
of  contact  P ;  from  A  and  B  let  fall  A  c  and  B  D 
perpendicular  to  D  p  c ;  then, 

angular  velocity  AP BD       BP 

~^ i       : —  Or  •  • « ( 1 ). 

angular  velocity  BP       AC       AP      vy 
This  result  may  be  expressed  in  language  as 
follows: — The  angular  velocities  of  the  wheels  are 
inversely  as  the  perpendiculars  let  fall  upon  the  com- 
mon tang  mt  from  the  centres  of  motion. 

Fig.  60.  Tiff.  51. 


73.  The  form  of  wheels,  represented  in  fig.  50, 
are   used  in  silk-mills,  and  in  the  Cometarium. 


WHEEL   WORK.  71 

The  curves  may  be  indefinitely  varied,  but  they 
must  always  be  constructed  to  answer  the  condi- 
tions explained  in  Art.  72. 

74.  Eoemer's  Wheels. — E  F  and  C  D  are  the  axea 
of  two  conical  wheels  or  bevel-wheels  K  and  G, 
having  their  vertices  turned  in  opposite  directions; 
the  teeth  of  K  are  formed  like  those  of  the  ordinary 
bevel-wheel;  but  the  teeth  on  G-  are  formed  by 
a  series  of  pins  e  k,  fixed  on  the  surface  of  the 
frustum  G.     By  varying  the  relative  position  of 
these  pins,  any  given  velocity  ratio  may  be  ob- 
tained. 

75.  Various  combinations  have  been  invented 
for  producing  a  varying  angular  velocity ;  such 
aa  the  eccentric  crown  wheel  and  broad  pinion, 
the  eccentric  spur-wheel  with  a  shifting  interme- 
diate wheel,  and  so  on. 


INTERMITTENT  AND  RECIPROCATING  MOTIONS  PRO- 
DUCED BY  WHEEL  WORK,  HAVING  ROLLING 
CONTACT. 

76.  The  following  is  an  example  of  an  inter- 
mittent motion  produced  by  the  continuous  mo- 
tion of  a  toothed  wheel. 

A  driving  wheel  A,  having  sunk  teeth  on  a  por- 
tion of  its  edge,  communicates  an  intermittent 
motion  to  the  wheel  B,  which  has  a  corresponding 
number  of  teeth  on  a  portion  of  its  edge.  The 


72  PRINCIPLES  OF  MECHANISM. 

portion  D  c  of  the  wheel  B,  being  a  plain  arc  of  a 
circle  described  from  A  as  centre,  allows  the  plain 

portion  of  the  wheel 
A  to  revolve  with- 
out any  interrup- 
tion. The  wheels 
are  brought  into 
gear  by  a  pin  p, 
fixed  to  the  wheel 
A,  and  a  GUIDE- 
PLATE  G  e,  fixed  to 

the  wheel  B.     Now  when  A  revolves,  in  the  direc- 
tion of  the  arrow,  the  plain  portion  of  its  edge 
runs  past  D  c  without  moving 

Fig.  63.  ,  ..  , 

the  wheel  B,  and  at  the  same 
time  keeps  it  from  shifting ; 
but  when  the  pin  p  comes  into 
contact  with  the  guide-plate,  the 
wheel  B  is  moved  round,  and 
the  teeth  D  E  engage  themselves 
with  the  teeth  on  B,  and  thus 
the  wheel  B  is  constrained  to 
make  a  revolution ;  it  then  re- 
mains at  rest  until  the  pin  p 
again  comes  round  to  meet  the 
guide-plate. 

77.  The  Rack  and  Pinion. — 
By  this  combination  a  circular 
reciprocating  motion  is  changed  into  a  reciprocat- 
ing rectilinear  one.  Teeth  are  cut  upon  the  edge 


WHEEL  WORK.  73 

of  the  straight  bars,  B  c  and  D  E,  so  as  to  work  with 
the  teeth  upon  the  pinion  A.  These  toothed  bars 
are  called  racks,  and  they  are  constrained  to  move 
in  rectilinear  paths  by  guides  or  rollers.  The 
racks  in  this  combination  move  in  opposite  direc- 
tions. 

78.  Fig.  53   represents   an  application  of  the 
double  rack,  for  converting  a  continuous  circular 

Kg.  54. 


motion  of  a  wheel,  A,  into  a  reciprocating  recti- 
linear motion,  given  to  the  frame  B  E. 

The  teeth  on  A  are  formed  by  pins  or  staves 
placed  about  one  quarter  round  the  face  of  the 
wheel ;  these  staves  act  alternately  upon  the  racks 
formed  on  the  upper  and  under  sides  of  the  frame. 
The  tooth  on  each  rack,  which  comes  first  into 
contact  with  the  stave  of  the  pinion,  is  made  longer 
than  the  others,  in  order  that  the  first  stave  should 
act  obliquely  upon  it,  thereby  tending  to  lessen 
the  shock.  In  this  figure  the  lower  stave  is  repre- 
sented as  leaving  the  last  rack  on  the  under  side, 
and  the  upper  stave  as  commencing  its  action  on 
the  elongated  tooth  of  the  upper  rack. 
7 


PRINCIPLES   OF   MECHANISM. 


V.   ON   SLIDING   PIECES,  PRODUCING   MOTION   BY 
SLIDING   CONTACT. 

The  Wedge  or  Movable  Inclined  Plane. 
79.  Let  A  B  c  be  a  movable  inclined  plane  or 
wedge,  sliding  along  the  smooth  surface  D  E,  by  a 
g.  55.  pressure    P  applied   to 

the  end  BC,  and  pro- 
ducing a  vertical  motion 
in  a  heavy  rod  G  PI  rest- 
ing on  the  plane  A  c,  and 
constrained  to  move  in 
a  straight  path  by  means 
of  guide  rollers.  The 
velocity  ratio  of  P  and  Px  will  be  constant,  being 
expressed  by  the  following  equality  : 

velocity  P        AB          length  of  the  wedge 


=  —  or 


velocity  P!       B  C  w"  thickness  of  the  wedge. 

To  transmit  motion  from  an  axis  A  D,  to  another 
axis  B  C,  parallel  to  it. 

80.  The  axis  A  D  carries  an  arm  A  E,  and  a  pin 
Fig.  se.  E  F,  which  enters  and 

slips  freely  in  a  slit 
made  in  the  arm  G  B 
attached  to  the  axis 
BC.  When  the  axis 
A  D  revolves,  it  com- 


municates a  rotation 
in  the  same  direction* 
to  the  other  axis  B  c, 


SLIDING  PIECES.  75 

but  with  a  varying  velocity  ratio,  for  the  pin  F 
continually  changes  its  distance  B  F  from  the  axis 
BC. 

When  the  distance  between  the  parallel  axes  is 
small,  and  the  axis  A  D  revolves  uniformly,  the 
angular  velocity  of  the  axis  B  c  varies,  very  nearly, 
inversely  as  the  distance,  B  F,  of  the  pin  from  this 
axis. 

T/ie  Eccentric  Wheel. 

81.    This  mechanism  is  usually  employed  to 
give  motion  to  the  slide-valve  of  the  steam  engine. 
Fig.  57.  In  fig.  57,  B  repre- 

sents the  axis  of 
the  eccentric 
wheel;  c  the  cen- 
tre of  the  circle; 
E  R  F  K  a  hoop 
which  embraces 
the  eccentric 
wheel  so  that  the  one  may  revolve  freely  within 
the  other  ;  E  F  D  a  frame  connecting  this  loop  with 
the  extremity  D  of  the  bent  lever  D  L  G,  turning 
on  the  fixed  centre  L.  Now  when  the  eccentric 
wheel  revolves  in  the  direction  of  the  arrow,  shown 
in  the  figure,  the  frame  with  the  pin  D  is  pushed 
to  the  right,  and  when  the  lob  side  of  the  eccen- 
tric has  passed  the  line  of  centres,  B  and  D,  the 
frame  with  the  pin  D  is  drawn  to  the  left,  and  so 
on.  Thus  the  continuous  rotation  of  the  axis  B 


76  PRINCIPLES   OF   MECHANISM. 

produces  a  reciprocating  circular  motion  in  the 
pin  D.  The  stroke  of  the  pin  D  will  be  equal  to 
twice  c  B,  or  double  the  eccentricity  of  the  wheel. 

Cambs,  Wipers,  and  Tappets. 

82.  Cambs  are  those  irregular  pieces  of  mechan- 
ism to  which  a  rotatory  motion  is  given  for  the 
purpose  of  producing,  by  sliding  contact,  recipro- 
cating motions  in  rods  and  levers. 

83.  In  fig.  58,  B  c  D  represents  the  camb,  turn- 
ing on  its  axis  A,  and  giving  a  reciprocating  recti- 
linear motion  to  the  heavy  rod 
E  F,  which  is  restrained  to  move 
in  its  rectilinear  path  by  the  guide 
rollers.     The  rotation  of  the  axis 
A  being  in  the  direction  of  the 
arrow,  the  rod  E  F  has  an  upward 
motion  until  the  extreme  point  B 
of  the  camb  comes  in  a  line  with 
the  rod,  then  the  portion  B  G  of 
the  carnb  allows  the  rod  to  fall, 
by  its  own  weight,  or  by  the  ac- 
tion of  a  spring,  until  the  point  G  comes  in  a  line 
with  the  rod,  and  so  on ;  thus  one  revolution  of 
the  camb,  here  presented,  will  cause  the  rod  to 
make  three  upward  and  three  downward  strokes. 
By  varying  the  curve  of  the  camb,  any  law  of 
motion  may  be  given  to  the  rod. 

84.  In  fig.  59,  the  pin  E  of  the  rod  is  made  to 
traverse  a  groove  E  G  D,  cut  in  the  camb  plate,  so 


SLIDING   PIECES. 


77 


Fie-. 


Fig.  60. 


that  the  pressure  of  the  camb  upon  the  pin  pro- 
duces the  downward  stroke  of 
the  rod  as  well  as  its  upward 
stroke.  In  this  case  the  rod 
will  only  make  one  upward  and 
one  downward  stroke  in  every 
revolution  of  the  camb  plate. 
The  length  of  the  stroke  of  the 
rod  will  be  equal  to  the  differ- 
ence between  AD  and  AG,  where 
D  is  the  point  in  the  groove 
furthest  from  the  centre  A,  and 
G  is  the  point  nearest  to  it. 

85.  To  find  the  curve  forming  the  groove  of  a  camb, 
so   that    the   velocity 
ratio  of  the  rod  and 
the  axis  of  the  camb 
may  be  constant. 

Let  A  be  the  cen- 
tre of  the  camb,  and 
c  A  B  Q  the  direction 
of  the  rod.  From  A 
as  a  centre,  with  any 
convenient  distance 
A  c,  describe  the  cir- 
cle c  E  D  B  N.  On 
B  A  take  B  a  equal 
to  the  length  of  the 
stroke  of  the  rod; 
divide  it  into  any 
7* 


78  PRINCIPLES   OF   MECHAXISM. 

convenient  number  of  equal  parts,  say  five,  in  the 
points,  b,  c,  d,  e,  and  divide  the  semicircle  B  D  E  F  G 
into  the  same  number  of  equal  parts  by  the  radial 
lines,  A  D,  A  E,  A  F.  From  A  as  a  centre,  with  A  b, 
A  c,  A  d,  A  e,  as  radii,  describe  circles  cutting  A  D, 
A  E,  &c.,  respectively  in  the  points  g,  k,  I,  m :  then 
through  these  points  draw  the  curve  a  g  k  I  m  c ; 
and  similarly  in  the  semicircle  B  N  c  draw  the 
other  curve  a  n  p  c. 

All  lines  drawn  through  the  centre  A  of  this 
curve  are  equal ;  thus  ac  =  ln  =  gp=  &c. 
Hence  if  the  rod  had  two  pins  placed  at  a  and  c, 
the  camb  would  revolve  between  them,  and  would 
cause  the  rod  to  make  a  downward  as  well  as  an 
upward  stroke.  This  curve  is  the  spiral  of  Archi- 
medes. 

By  dividing  the  line  B  a  into  parts  having  a 

Fig.  61. 


varying  ratio  to.  one  another,  any  proposed  law  of 
velocity  may  be  given  to  the  rod. 


SLIDING   PIECES. 


79 


Fig.  62. 


86.  In  fig.  61,  the  continuous  rotation  of  the 
camb  A  E  c,  revolving  on  the  axis  A,  gives  an 
oscillating  motion  to  the  rod  or  lever  F  a,  turning 
on  the  centre  F.     In  one  revolution  of  the  camb 
the  rod  makes  a  double  oscillation  in  the  arc  a  «L 

87.  Wipers. — When  the  rod  is  to  receive  a  series 
of  lifts  with  intervals  of 

rest,  the  camb  is  made 
into  the  form  of  project- 
ing teeth,  which  are  com- 
monly called  Wipers  or 
Tappets. 

88.  In  fig.  62,  the  re- 
volving cylinder  c  has  five 

wipers  upon  its  circumference,  which  give  five 
downward  strokes  to  the  hammer,  H,  placed  at  the 


Pig.  63. 


Fig.  64. 


extremity  of  the  lever  A  H,  in  each  revolution  of 
the  cylinder. 


80  PRINCIPLES   OF   MECHANISM. 

89.  In  fig.  63,  two  tappets,  upon  the  revolving 
cylinder  c,  give  two  downward  strokes  to  the 
heavy  bar  or  stamper  A  B,  in  each  revolution  of 
the  cylinder.     In  this  case  the  bar  A  B  is  con- 
strained to  move  in  a  rectilinear  path  by  means 
of  guide  rollers. 

90.  In  fig.  64,  a  single  wiper  on  the  cylinder  c 
gives  an  intermittent  rotation  to  the  ratchet  wheel 
A  with    its  detent  D.     At   each   revolution  of  c 
only  one  tooth  in  A  is  moved  round,  so  that  for 
the   greater   portion  of  the  revolution  of  c  the 
wheel  B  is  at  rest. 

91.  In  fig.  65,  the  continuous  rotation  of  three 

Fig.  6fl. 


wipers  a,b,  c,  communicates  a  reciprocating  recti- 
linear motion  to  the  frame  A  B  c  D.  The  wiper  a 
is  engaged  with  the  pallet  e,  and  at  the  instant 
of  disengagement  the  wiper  b  becomes  engaged 
with  the  pallet  g,  and  then  the  frame  starts  its 
motion  in  a  direction  contrary  to  that  of  the  ar- 
rows ;  and  so  on. 

The  Swash  Plate. 

92.  By  this   mechanism,  the  continuous   rota- 
tion of  an   axis   produces  a  reciprocating   recti- 


SLIDING   PIECES.  81 

linear   motion   in  a  rod,  in   the   direction  of  its 
length. 

Here  c  E  represents  the  revolving  axis,  to  the 
top  of  which  is  fixed  the  inclined  circular  flat 
plate  A  B,  called  the  swash  plate ;  A  D  F  the  rod 
to  which  a  reciprocating  motion  is  given  in  the 
the  direction  of  its  length,  having 
a  frictional  wheel  A  at  its  lower 
extremity  resting  on  the  swash 
plate.  This  rod  is  kept  in  contact 
with  the  plate  by  its  own  weight, 
or,  if  this  be  not  sufficient,  by 
means  of  a  spring.  Now  as  the 
swash  plate  turns  round,  the  rod 
A  F  is  alternately  raised  and  de- 
pressed, so  that  at  every  revolu- 
tion of  the  plate  the  rod  performs  an  upward 
and  a  downward  stroke.  Supposing  the  rod,  as 
represented  in  this  figure,  to  be  at  the  lowest 
point  of  its  stroke ;  from  c,  the  centre  of  motion 
of  the  piate,  let  fall  c  D  perpendicular  to  A  F  ;  then 
A  D  will  be  equal  to  half  the  stroke  of  the  rod. 
Moreover,  let  0  be  any  angle  moved  over  by  the 
axis,  and  let  h  be  the  corresponding  space  moved 
over  by  the  extremity  A  of  the  rod  ;  then 

h  =  A  D  +  (1  —  cos  e), 

which  gives  the  position  of  the  rod  at  any  point 
of  the  rotation  of  the  plate. 

93.  There  are  an  almost  endless  variety  of  com- 


82 


PRINCIPLES   OF   MECHANISM. 


binations  for  producing  reciprocating  motions  of 
this  kind,  by  means  of  sliding  contact. 

94.  In   fig.  67,  an   eccentric    revolving  pin  e, 
sliding  or  working  in  the  slit  of  the  arm  r  s  gives 
a  reciprocating  motion  to  the  rod  p  q  in  the  direc- 
tion of  its  length. 

95.  In  fig.  68,  the  same  effect  is  produced  by 


Fig.  67. 


Fig.  68. 


the  rotation  of  an  eccentric  wheel,  a  b,  on  its  axis 
a,  within  the  frame  c  D  F  E. 


SCREWS. 


96.   Construction  of  a  Helix  or  Screw. — Let  A  a  K 
be  a  cylinder,  and  A  D  E  a  piece  of  paper  cut  in  the 


Fig.  69 


SLIDING   PIECES.  83 

form  of  a  right  angled  triangle,  having  its  height 
D  E  equal  to  the  height  A  K  of  the  cylinder.  Now 
if  this  paper  be  wrapped  round  the  cylinder,  the 
slant  edge  A  E  of  the  paper  will  trace  the  helix  or 
screw  A  a  L  b  c  K  upon  the  cylinder.  If  A  B  =  B  0 
=  c  D  be  equal  to  the  circumference  of  the  cylin- 
der, the  edge  of  the  paper  will  form  four  convolu- 
tions, and  the  perpendiculars  BF  =  IG  =  HE will 
be  the  distance  between  the  threads  of  the  screw. 

97.  The  pitch  of  a  Screw  is  the  distance  B  F 
between  two  successive  convolutions.     If  t  =  B  F, 
the  distance   between  the  threads  of  the  screw, 
r  =  the  radius  of  the  cylinder,  0  =  angle  B  A  F  ; 
then 

_  2*r 
tan  e 

98.  We   may  also   conceive   the   helix  of  the 
screw  to  be  formed  by  the  compound  motion  of 
a  point.    Suppose  the  cylinder  to  rotate  uniformly 
upon  its  axis,  whilst  a  point  A  upon  its  surface  at 
the  same  time  moves  uniformly  in  the  direction  of 
its   length :    then. 

,  Fig.  70. 

with  this  compound 
motion,  the  point  A 
will  trace  the  helix 
of  a  screw. 

99.  Transmission 
of  motion  by  the  screw. 
— Let  e  a  n  c  m  g  be 


84  PRINCIPLES  OF  MECHANISM. 

a  spiral  groove  cut  upon  a  cylinder  ;  A  B  the  axis 
on  which  it  turns  ;  D  E  a  rod  parallel  to  the  axis 
A  B,  and  constrained  to  move  in  the  direction  of 
its  length  ;  e  a  tooth  attached  to  this  rod  fitting 
the  groove  of  the  screw.  Now  when  the  wheel  c 
is  turned  in  the  direction  of  the  arrow,  the  tooth 
with  the  rod  r>  E  will  be  moved  from  left  to  right 
in  the  direction  of  its  length,  that  is,  parallel  to 
the  axis  of  the  screw. 

The  velocity  ratio  of  the  wheel  c  and  the  rod 
D  E  will  be  constant,  for  we  have 

velocity  c         circum.  described  by  C 
velocity  CD~        pitch  of  the  screw 

If  R  be  the  radius  of  the  wheel  c,  r  =  the  radius 
cylinder  A  B,  v  =  velocity  circum.  c,  v  the  velo- 
city of  the  bar  D  E,  and  so  on  as  in  art.  97  ;  then 
the  above  equality  becomes  — 


v 


m. 

t          (  '' 


.'.-  =  -  tan  9  ...  (2); 
v        r  v  •" 


and  when  R  =  r,  then  — 


V 

-  =  tan  0  ...  (3): 

v 


that  is,  the  velocity  ratio  is  equal  to  the  tangent  of  the 
angle  which  the  thread  of  the  screw  makes  with  the 
sides  of  the  cylinder. 


SLIDING  PIECES, 


85 


100.  It  is  obvious  that  the  number  of  teeth  in 
the  bar  D  E  will  not  at  all  alter  its  motion. 

In  fig.  71,  the  screw  acts  upon  a  series  of  teeth 
Fig.  n. 


Fig.  72. 


upon  the  rack  D  E.  This  arrangement,  called  the 
rack  and  screw,  converts  a  circular  motion  into  a 
rectilinear  one. 

Solid  Screw  and  Nut. 

101.  In  general  the  piece  acted  upon  by  the 

screw  has  its  teeth,  or  rather  its 
threads,  formed  in  a  cavity  which 
embraces  the  whole  circumfer- 
ences of  the  screw,  and  the 
threads  of  the  one  exactly  fit- 
ting the  threads  of  the  other. 
This  modification  is  shown  in 
fig.  72,  where  N  is  the  hollow 
screw  fitting  the  threads  of  the  screw  s.  The  solid 
piece  s  is  called  the  male  screw,"  and  the  hollow 
piece  the  female  screw  or  nut. 

102.  Screws   are   either  left  handed,  or  right- 
handed,  according  to  the  direction  of  the  threads. 

103.  It  is  important  to  observe  that  the  follow- 
ing relations  of  motion  subsist  between  the  solid 

screw  and  the  nut : 
8 


86  PRINCIPLES   OF   MECHANISM. 

1.  When  the  nut  is  fixed,  the  solid  screw  will 
have  a  motion  in  the  direction  of  its  length,  upon 
being  turned  round. 

2.  If  the   nut  revolves,  without  having   any 
longitudinal  motion,  the  solid  screw  will  have  a 
motion  in  the  direction  of  its  length,  provided  it 
is  incapable  of  revolving. 

3.  If  the  solid  screw  revolves  without  having 
any  motion  in  the  direction  of  its  length,  the  nut 
will  have  a  longitudinal  motion,  provided  it  is  in- 
capable of  revolving. 

The  first  two  cases  are  exemplified  in  the  differ- 
ent forms  which  are  given  to  the  common  press, 
and  the  last  case  is  exemplified  in  the  construc- 
tion of  the  self-acting  slide  rest  of  the  lathe,  and 
in  other  kinds  of  mechanism. 

The  screw  is  usually  employed  for  producing 
very  slow  uniform  motions,  and  for  exerting 
great  pressure  through  a  limited  space. 

The  Common  Press. 

104.  In  fig.  73,  s  s  is  the  solid  screw,  N  the  nut, 
N  p  the  lever,  B  the  lower  press  board  which  is 
constrained  to  move  in  an  upward  direction  by 
means  of  the  guide  frame. 

Case  1.  In  this  case  the  nut  N  revolves,  but  does 
not  move  longitudinally,  but  the  screw  s  s  is  in- 
capable of  revolving.  Hence  the  press  board  B 
is  moved  upward  at  every  revolution  of  the  nut, 


SLIDING    PIECES. 


87 


over  a  space  equal  to  the  pitch  of  the  screw,  or  the 
distance  between  the  threads,  that  is, 

velo.  P  circum.  described  by  P 

velo.  B         distance  between  the  threads. 
Example. — Let  the  distance  between  the  threads 
=  J  in.,  the  length  of  the  lever  N  P  =  2  J  ft. ; 
required  the  velocity  ratio  of  the  point  P  and  the 
press-board  B. 

velo.  P      2  x  2£  x  12  x  3-1416 


velo.  B 


=  753-984 


Fig.  73  . 


That  is,  the  velocity  of  P  is  753'98-i  times  that 
of  B. 

Case  2.  In  this  case  N  is  a  perforated  cylinder 
forming  part  of  the 
solid  screw  s  s,  and 
therefore  turns  with  it 
on  a  pivot  which  works 
in  a  socket  placed  on 
the  under  side  of  the 
press  board  B ;  the 
piece  K  fixed  to  the 
frame  contains  the  hol- 
low or  female  screw ; 
so  that  the  solid  screw, 
s  s,  is  capable  of  re- 
volving and  of  moving 
longitudinally,  whilst 
the  nut  K  remains  ab- 
solutely fixed. 


88 


PRINCIPLES   OF   MECHANISM. 


Fig.  74. 


Compound  Screw. 

105.  This  mechanism  consists  of  two  screws  A 
and  D,  the  smaller  one  D 
working  within  the  larger 
one  A.  The  screw  A  works 
in  a  fixed  nut  or  female  screw 
at  K,  and  is  capable  of  revolv- 
ing and  moving  in  the  direc- 
tion of  its  length ;  the  small 
screw  D  is  incapable  of  re- 
volving, but  is  capable  of 
moving  in  the  direction  of  its 
length.  In  one  revolution  of 
the  lever  P,  the  screw  A  de- 
scends a  space  equal  to  the 
distance  between  its  threads, 
but  at  the  same  time  the  screw  D  enters  the  hollow 
screw  formed  in  A,  a  space  equal  to  the  distance 
between  the  threads  on  D,  so  that  the  extremity  B 
will  only  descend  a  space  equal  to  the  difference 
between  the  thickness  of  the  threads  on  A  and  the 
thickness  of  the  threads  on  B  ;  hence  we  have 

velo.  P  circum.  described  by  P 

velo.  B      dist.  bet.  th'ds  on  A  —  dist.  bet.  th'ds  on  D. 

If  the  length  of  the  lever  p  =  r,  the  pitch  of 
the  screw  A  =  t,  and  the  pitch  of  D  =  ^ ;  then 


velo.  P         2  ft  r 
velo.  B  ~  t  —  t 


...(1). 


SLIDTNG   PIECES. 


89 


n.,  ^  = 


Fig.  75. 


Example.  —  Let  r  =  5  ft.,   t  =  J 
the  a 

velo.r  =  2  x  5  x  12  x  3-1416 

velo.  B  |  —  f 

The  same  velocity  ratio  might  be  attained  by 
making  the  pitch  of  a  single  screw  A,  equal  to 
t  —  tt,  but  the  threads,  in  this  case,  might  be  too 
weak  to  stand  the  pressure  ;  hence  the  advantage 
of  the  compound  screw. 

The  Endless  Screw. 

106.  When  the  threads  or  teeth  of  a  revolving 
screw  are  made  to 
act  upon  the  teeth 
of  a  wheel,  as  in  fig. 
75,  the  mechanism 
is  called  the  endless 
screw.  Here,  each 
rotation  of  the  axis 
A  B  of  the  screw 
turns  round  one 
tooth  of  the  wheel 
c,  the  pitch  of  the 
screw  on  the  axis  A  B  being  equal  to  the  pitch  of 
the  teeth  on  the  wheel. 

If  Q  and  q  be  the  synchronal  rotations  of  the 
wheels  and  the  screw  respectively,  and  N  the  num- 
ber of  teeth  in  the  wheel  ;  then 


8* 


90  PRINCIPLES   OF    MECHANISM. 

If  N  =  40,  then  -  =  40  ;  that  is,  for  every  revo- 
lution performed  by  the  wheel  the  screw  will 
make  40. 

If  R,  r  be  the  respective  pitch-radii  of  the  wheel 
and  screw,  0  being,  as  before,  the  angle  which  the, 
thread  of  the  screw  makes  with  its  axis  ;  then 

-  =  -tan  0...(2). 
Q        r 

The  Differential  Screw. 

107.  A  D  is  an  axis  on  which  are  formed  two 
screws,  A  B  and  B  c,  whose  pitches  are  different. 

Fig.  76. 


The  screw  A  B  passes  through  a  fixed-  nut  or  fe- 
male screw  E,  whilst  B  c  passes  through  a  nut  N 
which  is  capable  of  moving  longitudinally,  but 
incapable  of  revolving  from  the  intervention  of 
the  guides. 

Let  the  screw  make  one  turn  so  as  to  move  the 
cylinder  from  right  to  left,  then  the  screw  A  B  will 


SLIDING   PIECES. 


91 


move  through  the  fixed  nut  E  a  space  equal  to  the 
distance  between  its  threads;  but,  at  the  same 
time,  the  screw  B  c  will  move  through  the  nut  N 
a  space  equal  to  the  thickness  of  the  threads  on 
B  c ;  so  that  the  nut  N  will  only  be  moved  through 
a  space  equal  to  the  difference  between  the  thick- 
ness of  the  threads  on  A  B  and  B  c,  that  is — 

In  one  revolution  of  A,  the  space  moved  over 
by  the  nut  N  =  pitch  screw  A  B  —  pitch  screw 
BC  =  <  —  ti,  where  t  is  put  for  the  pitch  of  the 
screw  A  B,  and  ^  for  that  of  B  c. 

If  t  =  ti,  then  nut  N  will  remain  at  rest. 

If  the  screw  A  B  be  right-handed,  and  B  c  left- 
handed  ;  then  t  -M i  will  be  the  space  moved  over 
by  the  nut  N  in  one  revolution  of  A. 

The  Archimedian  Screw  Creeper. 
108.  This  machine  is  used  for  conveying  corn 

Fig.  77. 

r»  c 


from  one  part  of  a  corn  mill  to  another.     It  con- 
sists of  a  wooden  cylindrical  trough,  A  B  c  D,  within 


92  PRINCIPLES   OF    MECHANISM. 

which  revolves  a  shaft,  E  F,  having  a  deep  spiral 
thread  formed  upon  its  surface.  The  corn  is 
dropped  in  at  one  extremity  of  the  trough  by  a 
hopper,  and  by  the  revolution  of  the  creeper  the 
corn  is  pushed  along  toward  the  other  extremity 
of  the  trough. 

Mechanism  for  cutting  Screws. 
109.  c  D  is  the  cylinder,  or  axis  on  which  the 


SLIDING   PIECES.  93 

screw  is  to  be  cut,  revolving  with  the  mandril  D 
of  the  lathe  ;  A  a  toothed  wheel  revolving  with 
the  axis  C  D,  and  giving  motion  to  the  toothed 
wheel  B,  round  its  axis  F  E,  on  which  is  cut  the 
parent  screw;  this  screw  gives  a  longitudinal 
motion  to  the  nut  N,  as  in  Case  3,  carrying  the 
sliding  table  or  saddle  upon  which  is  securely 
clamped  the  cutting  tool  p  intended  to  cut  the 
thread  of  the  screw  on  the  cylinder  C  D.  In  the 
place  of  the  wheels  A  B,  any  combination  of  wheels 
may  be  used  so  as  to  produce  any  relative  longi- 
tudinal velocity  to  the  cutting  tool  P,  and  thereby 
to  form  a  screw  of  any  given  pitch  on  C  D  with 
the  same  parent  screw  F  E. 

Let  n  =  the  no.  of  teeth  on  the  wheel  A,  nt  = 
the  no.  of  teeth  on  B,  t  =  the  pitch  of  the  screw 
on  CD,  ti  =  the  pitch  of  the  screw  on  F  E  ;  then 


which  expresses  the  pitch  of  the  screw  on  c  D. 
From  this  equality  we  get, 


that  is  to  say,  the  pitches  of  the  screws  are  in  the 
ratio  of  the  number  of  teeth  on  their  respective  wheels. 
If  HI  and  ti  be  constant,  then 

t  oo  n, 

that  is  to  say,  the  pitch  of  the  screw  on  C  D  varies 
with  t/ie  number  of  teeth  on  its  w/ieel  A. 


94  PKINCIPLES   OF   MECHANISM. 

Let  k  and  7^  be  the  number  of  threads  per  inch 
on  the  cylinders  C  D  and  F  E  respectively,  then 

1  ,  1 

£  =  t,  and  £  =  tlt 

and  eq.  (2)  becomes — 

•*-=-'. ..(3). 

ki       n 

Now,  let  there  be  an  intermediate  pinion  and 
wheel,  turning  on  the  same  axis,  placed  between 
A  and  B ;  and  let  the  pinion  (acted  upon  by  A) 
contain  eL  teeth,  and  the-  wheel  e, teeth  ;  then  the 
velocity  ratio  of  the  axis  F  E  will  be  increased  by 

e 
the  ratio  -.  and  hence  eq.  (3)  becomes — 


KI      ne 

Example. — Let  n  =  30,  n^  =  10,  ^  =  J  in. ;  re- 
quired t. 

n  30 

Here  by  eq.  (1),  t=~   .  ^  =  ----  x  |  =  1|  in. 
ni 

To  produce  a  changing  reciprocating  rectilinear  mo- 
tion by  a  combination  of  the  camb  and  screw. 

\  110.  E  F  is  a  conical  shaped  camb,  turning  on 
the  eccentric  axis  A  B,  on  which  is  cut  the .  screw 
K  B,  working  in  the  fixed  nut  or  hollow  screw  N ; 
D  c  a  rod  resting  on  the  camb,  constrained  to 


SLIDING   PIECES.  95 

move  in  the  direction  of  its  length,  and  to  which 
the  varying  reciprocat- 
ing motion  is  to  be 
given.  Here,  whilst 
the  camb  revolves,  it 
has  a  continuous  mo- 
tion in  the  direction  of 
the  axis  A  B,  so  that  the 
lower  extremity,  c,  of 
the  rod  D  c  describes  a 
spiral  or  screw  curve 

upon  the  cone  whose  pitch  is  equal  to  the  pitch 
of  the  screw  K  B.  The  effect  of  this  is  to  make  c  D 
reciprocate  in  its  path  in  such  a  manner  that  the 
stroke  in  one  direction  is  shorter  than  that  in  the 
opposite,  direction. 

To  produce  a  boring  motion  by  a  combination  of 
the  screw  and  toothed  wheels. 

111.  Here  it  is  required  to  produce  a  rapid  ro- 
tation combined  with  a  very  slow  motion  in  the 
direction  of  the  axis. 

The  screw  I  B  is  cut  upon  a  portion  of  the  re- 
volving axis  A  B  ;  this  screw  passes  through  a  nut 
K  capable  of  revolving  with  the  wheel  G-,  but  in- 
capable of  moving  in  the  direction  of  its  axis,  as 
in  Case  2,  page  87;  the  wheel  G  is  driven  by  the 
pinion  F  revolving  on  the  parallel  axis  DC;  E  is 
a  long  pinion,  turning  on  this  axis,  and  acting  on 
the  wheel  D,  which  transmits  a  rotatory  motioc 


96 


PRINCIPLES  OP   MECHANISM. 


to  the  screw  axis  A  B.  Now  the  rotation  of  C  D 
produces  a  rotatory  motion  in  the  axis  A  B,  and  at 
the  same  time  causes  it  to  advance,  in  the  direc- 


Fig.  80. 


tion  of  its  length,  with  a  velocity  determined  by 
the  following  formula. 

Let  Q,  Qu  £1  be  the  synchronal  rotations  of  "the 
axis  c  D,  the  nut  K  and  wheel  G,  and  the  wheel 
and  axis  A  B,  respectively  ;  N,  NI,  n,  HI,  the  number 
of  teeth  in  the  wheels  F,  G,  E,  L,  respectively ;  s 
the  space  moved  over  by  A  B  in  the  direction  of 
its  length  ,  and  t  =  the  pitch  of  the  screw  I  B. 

Now  Q!  rotations  of  the  nut  K  moves  the  screw 
A  B  through  a  space  equal  to  Qt  X  t ;  but  q^  rota- 
tions of  L  moves  the  screw  through  a  space,  in 
the  opposite  direction,  equal  to  qr  x  t ;  therefore 
in  Q  rotations  of  the  axis  c  D,  the  screw  A  B  will 
be  moved  through  a  space  equal  to  the  difference 
between  Q  X  t  and  qv  x  t,  that  is, 


SLIDING  PIECES.  97 

,        Qi         N  ft        n 

but  -  =  -  ,  and  -  =  -  ; 
Q        Ni          Q       in 


71  N 

Now  the  difference  ---  may  be  very  small 

Wi         N!        J 

as  compared  with  Q,  and  consequently  s  may  be 
made  as  small  as  we  please  as  compared  with  Q, 
which  is  the  condition  required  for  the  construc- 
tion of  a  boring  instrument.  The  boring  tool  is 
placed  upon  one  extremity  of  the  axis  A  B. 


MACHINERY  OF  TRANSMISSION. 


CHAPTER    II. 

ON   WHEELS   AND    PULLEYS. 

THE  elementary  principles  of  motion  by  rolling 
contact  and  by  wrapping  connectors  have  already 
been  explained,  so  that  in  the  present  chapter  we 
have  only  to  examine  in  detail  the  methods  of  ap- 
plying these  principles  and  their  respective  advan- 
tages, and  especially  the  mode  of  constructing 
wheels  in  gear,  so  that  the  resulting  motion  shall 
most  nearly  approach  the  condition  of  perfect  roll- 
ing contact. 

We  saw  in  the  preliminary  chapter  that  there 
were  two  methods  of  transmitting  power  through 
trains  of  wheel  work,  the  first  being  through  the 
agency  of  wrapping  connectors,  and  the  second  by 
rolling  contact. 

Wrapping  connectors. — Considerable  difference 
of  opinion  exists  as  to  the  best  and  most  effective 
principle  of  conveying  motion  from  the  source  of 
power  to  the  machinery  of  a  mill.  The  Americans 
prefer  leather  straps,*  and  large  pulleys  or  riggers. 

*  I  have  selected  the  word  strap,  instead  of  belts  or  bands, 
as  a  term  more  generally  applied  to  wrapping  connectors 
iu  the  northern  districts. 

(99) 


100  MACHINERY  OF  TRANSMISSION. 

In  this  country,  and  especially  in  the  manufactur- 
ing districts,  toothed  wheels  are  almost  universally 
employed.  In  some  parts  of  the  South,  and  in 
London,  straps  are  extensively  used ;  but  in  Lan- 
cashire and  Yorkshire,  where  mill- work  is  carried 
out  on  a  far  larger  scale,  gearing  and  light  shafts 
at  high  velocities  have  the  preference.  Naturally, 
I  am  of  the  opinion  that  the  North  is  right  in  this 
matter,  and  that  consistently,  as  I  was  to  a  great 
extent  the  first  to  introduce  that  new  system  of 
gearing  which  is  now  general  throughout  the 
country,  and  to  which  I  have  never  heard  any 
serious  objection.  I  have  been  convinced  by  a 
long  experience  that  there  is  less  loss  of  power 
through  the  friction  of  the  journals,  in  the  case  of 
geared  wheel  work,  than  when  straps  are  employed 
for  the  transmission  of  motive  power.  Carefully 
conducted  experiments  confirm  this  view,  and  it 
is  therefore  evident  which  mode  of  transmission 
is,  as  a  general  rule,  to  be  preferred. 

There  are  certain  cases  in  which  it  is  more  con- 
venient to  use  straps  instead  of  gearing.  With 
small  engines  driving  saw-mills,  and  some  other 
machinery  where  the  action  is  irregular,  the  strap 
is  superior  to  wheel  work,  because  it  lessens  the 
shocks  incidental  to  these  descriptions  of  work. 
So,  also,  when  the  motive  power  has  been  con- 
veyed by  wheel  work  and  shafting  to  the  various 
floors  of  a  mill,  it  is  best  distributed  to  the  ma- 
chines by  means  of  straps. 


WHEELS   AND   PULLEYS.  101 

In  some  of  the  American  cotton  factories,  bow- 
ever,  there  is  an  immense  drum  on  the  first  mo- 
tion, with  belts  or  straps  from  two  to  three  feet 
wide,  transmitting  the  power  to  various  lines  of 
shafting,  and  these  in  turn  through  other  pulleys 
and  straps,  giving  motion  to  the  machinery.  From 
this  description  it  will  be  seen  that  the  whole  of 
the  mill  is  driven  by  straps  alone,  without  the 
intervention  of  gearing. 

The  advantages  of  straps  are,  the  smoothness 
and  noiselessness  of  the  motion.  Their  disadvan- 
tages are,  cumbrousness,  the  expense  of  their  re- 
newal, and  the  necessity  for  frequent  repairs.  They 
are  inapplicable  in  cases  where  the  motion  must 
be  transmitted  in  a  constant  ratio,  because,  as  the 
straps  wear  slack,  they  tend  to  slip  over  the  pul- 
leys, and  thus  lose  time.  In  other  cases,  as  has 
been  observed,  this  slipping  becomes  an  advan- 
tage, as  it  reduces  the  shock  of  sudden  strains, 
and  lessens  the  danger  of  breaking  the  machinery. 

Very  various  materials  are  employed  for  straps, 
the  most  serviceable  of  all  being  leather  spliced 
with  thongs  of  hide  or  cement.  Gutta  percha  has 
been  employed  with  the  advantage  of  dispensing 
with  joints,  but  it  is  affected  by  changes  of  tem- 
perature, and  it  stretches  under  great  strains. 
Flat  straps  are  almost  universally  employed,  in 
consequence  of  the  property  they  possess  of  main- 
taining their  position  on  pulleys,  the  edge  of 

which  is  slightly  convex  (fig.  81).     Bound  belts 
9* 


102  MACHINERY   OF  TRANSMISSION1. 

of  catgut  or  hemp  are  sometimes  used,  running  in 

grooves,  which  are  better  made  of  a       F1    81 

triangular  than  a  circular  section — so 

that  the  belt  touches  the  pulley  in  two 

lines  only,  tangential  to  the  sides  of 

the  groove ;  in  this  case  the  friction 

of  the  belt  is  increased  in  proportion 

to  the  decrease  of  the  angle  of  the 

groove. 

The  strength  of  straps  must  be  determined  by 
the  work  they  have  to  transmit.  Let  a  strap  trans- 
mit a  force  of  n  horses'  power  at  a  velocity  of  v 
feet  per  minute,  then  the  tension  on  the  driving 

33000  n  „ 
side  of  the  belt  is  —     —  Ibs.  independent  of  the 

initial  tension  producing  adhesion  between  the 
belt  and  pulley.  For  example,  let  v  be  314'16 
feet  per  minute,  or  the  velocity  of  a  24-inch  pulley 
at  50  revolutions  per  minute,  and  let  3  horses' 

33000  x  3 
power  be  transmitted ;  then — oiT^r/i —  =312  Ibs., 

the  strain  on  the  pulley  due  to  the  force  trans- 
mitted. 

The  following  table  has  been  given  for  deter- 
mining the  least  width  of  straps  for  transmitting 
various  amounts  of  work  over  different  pulleys. 
The  velocity  of  the  belt  is  assumed  to  be  between 
25  and  30  feet  per  second,  and  the  widths  of  the 
belts  are  given  in  inches.  With  greater  velocities 
the  breadth  may  be  proportiouably  decreased. 


WHEELS   AND   PULLEYS. 


103 


TABLE  I. — APPROXIMATE  WIDTHS  OF  LEATHER  STRAPS,  IN 
INCHES,  NECESSARY  TO  TRANSMIT  ANY  NUMBER  OF  HORSES' 
POWER. 


Smallest  Diameter  of  Pulley  in  Feet. 

Horses'  Power. 

1 

2 

3 

4 

6 

6 

7 

8 

10 

l 

3-6 

1-8 

1-2 

2 

7-2 

3-6 

2-4 

1-8 

1-4 









3 

10-8 

6-4 

3-6 

2-7 

2-1 

18 

1-6 





4 

14-4 

7-2 

4-8 

3'6 

4-8 

2-4 

2-0 

1-8 

1-4 

'  6 

18-0 

9-0 

60 

4-6 

3-6 

3-0 

2-6 

22 

1-8 

7 

25-2 

12-6 

8-4 

6-3 

6-4 

4-2 

3-5 

3-7 

2-6 

10 

36-0 

180 

12-0 

9-0 

7-2 

6-0 

5-1 

4-6 

36 

12 

43-2 

21-6 

14-4 

10-8 

8-6 

7-2 

6-1 

64 

4-3 

14 

25-2 

16-8 

126 

lO'O 

8'4 

7-1 

63 

6-0 

16 



28-8 

19-2 

14-4 

11-5 

96 

8'2 

7-2 

6-7 

18 



32-4 

21-6 

16-2 

12-9 

10-8 

9-2 

8-1 

6-4 

20 



36-0 

24-0 

18-0 

14-4 

120 

10-2 

90 

7-2 

25 



45-0 

30-0 

22-6 

18-0 

15-0 

12-8 

11-2 

9-0 

30 



360 

27-0 

21-0 

18-0 

16-0 

13-0 

10-0 

40 

___ 



48-0 

36-0 

28-0 

240 

200 

18-0 

14-0 

60 

___ 



^__ 

460 

36-0 

30-0 

25-0 

22-0 

18-0 

60 









43-0 

36-0 

30-0 

27-0 

21-0 

70 





___ 



__ 

42-0 

35-0 

31-0 

25-0 

80 

___ 

_^_ 

_^_ 



^_ 

—  _ 

41-0 

36-0 

28-0 

100 













61-0 

450 

360 

Toothed  Wheels. — The  second  method  of  com- 
municating motion  is  by  rolling  contact,  as  ex- 
plained in  the  preliminary  chapter.*  But,  in 
practice,  the  adhesion  between  the  surfaces  is 
seldom  sufficient  to  communicate  the  necessary 
power,  and  hence  various  contrivances — such  as 
the  wheel  and  trundle,  and  toothed  wheels — have 
been  substituted.  The  general  equations  for  velo- 
city, ratio,  etc.,  are  the  same  as  if  the  wheels  rolled 
on  each  other  at  the  pitch  circles,  but  in  fact  each 
tooth  slides  upon  its  fellow.  The  determination 

*  See  page  56. 


104 


MACHINERY   OF   TRANSMISSION. 


of  the  best  forms  of  these  teeth  so  that  the  friction 
shall  be  a  minimum  and  the  motion  uniform,  is 
one  of  the  most  important  contributions  of  applied 
mathematics  to  practical  engineering. 

Of  the  introduction  of  toothed  wheels  and 
toothed  gearing,  we  know  very  little.  Hero  of 
Alexandria,  who  wrote  two  centuries  before  our 
ara,  speaks  of  toothed  wheels  and  toothed  bars  in 
a  way  which  seems  to  indicate  that  he  was  not 
altogether  ignorant  of  this  method  of  transmitting 
motion.  Later  forms  are  figured  in  great  variety 
in  the  different  collections  of  mechanical  appli- 
ances of  the  sixteenth  and  seventeenth  centuries. 

Spur  gearing  is  employed  where  the  axes  on 
which  the  wheels  are  placed  are  parallel  to  one 
another.  The  smaller  wheel 
in  a  combination  of  this  sort 
is  termed  the  pinion.  An- 
nexed (fig.  82)  is  a  pinion 
from  Eamelli  (A.  D.  1588), 
which  from  its  form,  may  be 
surmised  to  be  of  metal.  The 
principle  on  which  spur  gear- 
ing is  constructed  is  primarily  « 
the  communication  of  motion  through  the  rolling 
>f  two  cylinders  on  one  another.  The  teeth  are 
mtroduced  to  prevent  slipping,  and  thus  to  insure 
the  regular  communication  of  the  motive  power. 

In  the  older  wooden  wheels,  the  teeth  were 
usually  formed  of  hard  wood,  and  driven  into 


Fig.  82. 


WHEELS   AND   PULLEYS.  104 

mortises  on  the  periphery  of  a  wooden  wheel. 
The  pinions  were  generally  replaced  by  trundles, 
in  which  cylindrical  staves,  fixed  at  equal  dis- 
tances round  the  periphery  of  two  discs,  were 
driven  by  the  teeth  of  the  wheel. 

The  mortise  wheels  are  still  retained  in  countries 
where  iron  is  expensive,  and  even  in  this  country 
they  are  employed  in  a  modified  form.  Iron 
pinions,  with  wooden  cogs  fixed  in  the  periphery, 
are  used  to  receive  the  motion  from  the  fly-wheels 
of  engines,  with  a  view  to  reduce  the  noise  and  to 
increase  the  smothness  of  the  motion ;  and  many 
millwrights  prefer,  in  all  cases  where  large  wheels 
are  required  to  run  at  high  velocities,  to  make  one 
of  them  a  mortise-wheel,  with  wooden  cogs. 

There  does  not  appear  to  have  been  much  im- 
provement in  the  construction  of  wood  and  iron 
gear  since  it  was  first  introduced  by  Mr.  Eennie  ; 
the  only  exception  being  the  introduction  of  a 
machine  for  cutting  out  the  form  of  the  teeth,* 
which  in  those  days  was  done  by  hand,  with  keys 
or  wooden  wedges  fitting  into  dovetails  in  the 
'  shanks '  of  the  cogs,  as  shown  at  a,  fig.  83,  on  the 

*  Mr.  Smiles  states,  in  his  '  Lives  of  the  Engineers,'  that 
Brindley,  more  than  a  century  ago,  invented  machinery 
for  the  manufacture  of  tooth  and  pinion  wheels,  'a  thing,' 
as  stated  by  the  author,  '  that  had  not  before  been  attempt- 
ed, all  such  wheels  having,  until  then,  been  cut  by  hand, 
at  great  labor  and  cost.' 


106 


MACHINERY   OF   TRANSMISSION. 


Fig.  83. 


concave    side    of 

the  rim  ;  now  they 

are  made  with  an 

iron     pin    driven 

through  the    cog, 

close  to  the   rim, 

as  at  b.    The  iron 

pinion  or  wheel  intended  to  work  in  contact  with 

the  wood  teeth  was,  up  to  a  recent  date,  turned 

and  carefully  divided  to  the  epicycloidal  form, 

Fig.  84. 


WHEELS  AND  PULLEYS.  107 

and  then^  chipped  and  filed  with  great  exactitude, 
in  order  to  fit  accurately  into  the  wooden  teeth  of 
the  driving  wheel.  In  all  the  corn  mills  of  the 
present  day,  and  where  great  speed  is  required, 
the  same  attention  to  accuracy  is  observed  in 
wood  and  iron  gear  as  in  former  times. 

The  greatest  advance  in  the  application  of  gear- 
ing resulted  from  the  introduction,  at  the  end  of 
the  eighteenth  century,  of  cast  iron  in  place  of 
wood.  The  credit  of  the  introduction  of  this  ma- 
terial is  usually  given  to  Smeaton,  who  began  to 
use  cast  iron  in  the  construction  of  the  Carron 
Kolling  Mill,  in  1769.  But  the  late  Mr.  John 
Eennie,  when  at  Boulton  and  Watt's,  in  1784,  was 
probably  the  first  to  carry  the  use  of  cast  iron  into 
all  the  details  of  mill  work.  Figs.  84,  85  are 
copied  from  the  original  designs  for  the  Soho 

Boiling  Mill,  dated 
1785.  But  the  Al- 
bion Corn  Mills, 
built  about  the  same 
time  (1784-5),  may 
be  considered  as  the 
earliest  instance  of 
the  entire  replacing 
of  wood  by  cast 
iron  for  the  bevel  and  spur  wheels  and  shafts. 
This  was  effected  by  the  same  distinguished  en- 
gineer. 

Where  the  shafts  of  the  wheel  and  pinion  are 


108  MACHINERY   OF   TRANSMISSION. 

not  parallel  to  each  other,  various  forms  of  conical 
trundles  and  bevel  wheels,  are  employed.  The 
simplest  plan  is  probably  the  face  wheel  and  trun- 
dle shown  in  fig.  86,  which  have  been  employed 
from  a  very  early  period,  and  which,  if  made  of 
metal,  take  the  form  of  the  crown  wheel  and  pin- 
ion, fig.  87.  Where  the  axes  are  not  at  right 
angles,  conical  trundles  have  been  used,  one  of 
which  is  figured  in  Bessoni  (A.  D.  1578.) 

The  most  perfect  arrangement,  however,  is  that 
in  which  two  wheels  called 
bevel  wheels  are  employed 
constructed  in  the  form  of 
frusta  of  cones.  These 
were  not  introduced  till 
the  middle  of  the  last  cen- 
tury, the  principles  of  the 
construction  of  the  teeth 
being  due  to  Camus  (A.D. 
1752).  Fig.  88  shows  a  bevel  wheel  designed  for 
the  Boiling  Mill  at  Soho,  by  the  late  Mr.  Eennie, 
in  1785* 

*  It  is  evident  from  the  shape  of  the  eye  of  these  wheels, 
figs.  84,  85,  and  88,  that  they  were  intended  for  wooden 
shafts,  and  that  cast  iron  had  not  been  in  use  much  before 
that  time.  At  an  earlier  period,  Mr.  W.  Murdock,  of  Soho, 
had  a  cast  iron  bevel  wheel,  which  was  considered  the  first 
introduced  into  Scotland,  many  years  previous  to  the  above 
date.  Mr.  Smeaton  also  had  introduced  iron  wheels  at  Car- 
ron  in  1754,  and  afterwards  at  a  mill  at  Belper,  in  Derby- 
shire. (See  Siniles's  "Life  of  Rennie,"  page  138. ) 


WHEELS  AND  PULLEYS.  109 

The  smoothness  and  economy  of  wheel  work  de- 
pend entirely  upon  the  accuracy  of  the  curvature 
of  the  individual  teeth  which  gear  with  one  an- 
other. Two  chief  defects  result  from  imperfec- 
tons  in  their  construction  :  first,  the  motion  con?- 

Fig.  88. 


municated  to  the  driven  wheel  is  irregular,  in- 
creasing and  diminishing  alternately  as  each  tooth 
passes  the  line  of  centres ;  and,  second,  there  is  an 
unnecessary  friction  between  the  teeth  in  gear, 
resulting  not  only  in  loss  of  power,  but  also 
causing  a  great  and  destructive  wear  in  the  teeth 
10 


110  MACHINERY  OF  TRANSMISSION. 

and  journals.  These  defects  can  only  be  avoided 
by  reducing,  as  far  as  praticable,  the  size  of  the 
teeth,  and  by  the  adoption  of  true  principles  in 
setting  out  their  curvature  in  the  original  model. 

To  the  first  cause  alone  a  large  part  of  the  per- 
fect action  of  modern  machinery  of  transmission 
is  to  be  attributed;  but  there  is  moreover  no 
doubt  that,  in  practice,  even  where  true  principles 
have  not  been  adopted,  a  considerable  approach 
has  been  made  to  such  forms  as  theory  requires. 
Now,  with  certain  limitations,  it  is  known  that  if 
any  form  of  tooth  be  taken  for  one  wheel,  there 
can  be  found  another  tooth  which  will  work  cor- 
rectly with  it.  Bat  there  are  certain  forms  which, 
being  susceptible  of  accurate  mathematical  deter- 
mination, are  most  convenient  for  the  purpose. 
Camus,  in  1752,  was  the  first  to  work  out  the  pro- 
perties of  epicycloidal  and  hypocycloidal  curves 
when  employed  in  the  construction  of  the  teeth  of 
spur  and  bevel  gearing.  De  la  Hire  adopted  the 
same  form.  Euler,  in  1760,  and  Kaestner,  in  1771, 
investigated  in  a  similar  manner  the  properties 
of  the  involute.  Since  their  time,  Ferguson,  Bu- 
chanan, Hawkins,  Rennie,  and  Airy,  have  all  con- 
tributed to  perfecting  the  mathematical  theory. 
And  Professor  Willis,  amongst  other  important 
additions,  has  shown  how  a  close  approximation 
to  a  true  form  may  be  made  by  the  adoption  of  a 
system  of  circular  arcs.  I 

From  1788,  when  Rennie  completed  the  Albion 


WHEELS  AND  PULLEYS.  Ill 

Mills,  to  the  present  time,  wood  and  iron  gear 
have  been  in  general  use  for  high  velocities,  and 
for  every  description  of  machinery  where  smooth- 
ness and  accuracy  of  motion  were  required.  Mr. 
Rennie  was  the  first  to  introduce  this  system ;  and 
in  most  cases  he  made  the  driver,  or  large  wheel, 
with  wood  cogs,  and  the  driven,  or  pinion,  of  iron 
"  chipped  and  turned  " — that  is,  every  tooth  of  the 
iron  wheel  was  carefully  divided  in  the  pitch, 
having  first  been  turned  on  the  fane  and  the  ends 
of  the  teeth,  and  drawn  to  the  epicycloidal  form. 
They  were  then  chipped  with  the  hammer  and 
chisel,  and  accurately  filed  to  the  required  dimen- 
sions and  forms.  The  same  process  was  applied 
to  the  wooden  teeth ;  and  these  wheels,  when  duly 
prepared,  were  keyed  on  their  respective  shafts, 
and  securely  fixed  in  contact  in  the  mill. 

The  chipping  and  filing  process  has  of  late 
years  been  superseded  by  a  cutting  machine, 
which  effects  the  same  purpose,  with  less  risk  of 
error  ;  and  the  good  old  system  of  a  penny  an  inch, 
as  practised  in  Eennie's  time,  has  been  exploded, 
much  to  the  discomfiture  of  the  old  millwrights, 
who  adhere  with  great  tenacity  to  the  hammer 
and  chisel.  Fig.  89  shows  the  cutting  machine  as 
constructed  by  Messrs.  Peter  Fairbairn  and  Co.,  of 
Leeds. 

The  object  of  this  machine  is  not  only  to  pitch 
and  trim  the  teeth  of  a  large  spur  or  other  wheel, 


112  MACHINERY   OF  TRANSMISSION. 

Fig.  89. 


WHEELS   AND   PULLEYS.  113 

but  to  turn  the  face  and  sides  of  the  segments 
previously,  when  bolted  to  the  arms. 

When  used  as  a  lathe  for  turning,  the  parts  in 
use  are  as  follows  :  B  is  a  large  headstock,  carry- 
ing a  hollow  spindle  (C),  through  which  is  inserted 
a  mandrill  upon  which  the  wheel  to  be  cut  and 
turned  is  keyed.  Provision  is  made  for  carrying 
the  other  end  of  this  mandrill  by  a  loose  fixing. 
The  hollow  spindle  is  driven  (with  the  wheel  upon 
it)  by  a  worm  wheel  (J)  which  is  made  to  run 
lootse  on  the  spindle,  but  which  is  now  by  a  lock 
bolt  connected  to  the  larger  worm  wheel  or  divid- 
ing wheel  (E),  the  worm  of  which  is  now  thrown 
out,  and  which  is  keyed  firmly  on  the  spindle. 
The  necessary  speeds  are  given  by  the  five-speed 
cone  and  mitre  gear.  The  tool  for  turning  is  held 
in  an  ordinary  slide  rest,  which  moves  transver- 
sally  on  a  saddle,  which  slides  and  is  fastened  in 
the  T  groves  of  two  strong  beds  (A),  firmly  se- 
cured to  masonry,  and  between  which  the  wheel 
revolves. 

When  used  for  pitching  and  trimming,  the  lock 
bolt  connecting  the  two  worm  wheels  is  removed, 
and  the  pitch  is  given  by  the  train  of  change 
wheels  and  division  plate  (A).  The  place  of  the 
slide  rest  is  now  taken  by  a  headstock  carrying 
two  cutters,  one  for  roughing,  and  the  other  for 
finishing. 

The  finishing  rose-cutter  is  the  counterpart  of 
10* 


114 


MACHINERY   OF  TRANSMISSION. 


the  space  between  the  teeth,  and  is  transversed 
across,  making  both  sides  of  the  tooth  alike. 

The  remainder  of  the  arrangement  will  be 
obvious  from  the  sketch.  The  same  machine  can 
be  also  readily  arranged  for  cutting  worm-wheel 
teeth,  or  for  bevel  gear. 

The  best  form  which  can  be  given  to  the  teeth 
of  wheels  is  that  which  will  cause  them  to  be 
always,  in  regard  to  the  power  they  mutually 
exert,  in  equally  favorable  situations,  and,  con- 
sequently, will  give  the  machine  the  property  of 
being  moved  uniformly  by  a  power  constantly 
equal.  This  would  be  accomplished  by  simple 
rolling  contact,  which  corresponds  with  the  case 
in  which  the  teeth  are  infinitely  small. 

Definitions. 

1.  Spur  gearing  is  that  in  which  the  pitch  lines 
of  the  driving  and  driven  wheel  are  in  the  same 
plane  (fig.  90). 

Fig.  90.  Fig.  91. 


2.  Bevel  gearing  is  that  in  which  the  planes  of 
the  pitch  lines  of  the  driving  and  driven  wheel 


WHEELS  AND   PULLEYS.  115 

are  inclined  to  each  other.     In  practice,  they  are 
in  most  cases  at  right  angles  (fig.  91). 

3.  Of  two  wheels  in  gear,  the  lesser  is  called 
the  pinion. 

4.  When   two  wheels  are  in  gear,  a  straight 
line  joining   their  centres  is  called  the  line  of 
centres. 

5.  If  the  line  of  centres  be  divided  into  two 
parts,  proportionally  to  the  number  of  teeth  in 
the  wheel  and  pinion,  these  parts  are  called  the 
proportional  or  primitive  radii  of  the  wheel  and 
pinion. 

6.  The  radii  of  the  circles  which  limit  the  ex- 
tremities of  the  teeth  are  called  the  true  radii. 

7.  If,  from  the  centres  of  the  wheel  and  pinion, 
circles  be  drawn  with  radii  equal  to  the  primitive 
radii,  so  that  they  touch  one  another  in  the  line 
of  centres,  the  circles  are  called  the  pitch  lines  of 
the  wheel  and  pinion  respectively. 

8.  The   acting   surface  of  a  tooth,  projecting 
beyond  the  pitch  circle,  is  called  its  face ;  thaf 
enclosed  within  the  pitch  circle,  its  flank. 

9.  The  pitch  of  a  wheel  is  the  distance  measured 
along  the  pitch  circle  from  the  face  of  one  tooth 
to  the  corresponding  face  of  the  next;  it  includes, 
therefore,  the  breadth  of  a  tooth  and  space.     For 
two  wheels  to  work  in  gear,  the  pitch  must  be 
the  same  in  each. 

10.  Backs  are  toothed  bars  in  which  the  pitc1* 
line  is  a  straight  line. 


116  MACHINERY   OF   TRANSMISSION. 

11.  In  annular  wheels  the  teeth  are  cut  on  the 
internal  edge  of  an  annulus,  or 
ring  (fig.  92.) 

In  fig.  93,  B  F  is  the  line  of 
centres ;  F  A,  A  B,  the  primitive 
radii  of  the  wheel  and  pinion 
respectively  ;  A  K  L  and  A  M  N 
the  pitch  lines ;  K  L  and  M  N,  the 
pitch ;  p  L,  the  face ;  and  Q  L  the  flank,  of  the 
tooth. 

Fig.  93. 


The  pitch  of  Wheels. 
We  have  seen  that  the  pitch  of  a  wheel  is  the 


WHEELS  AND   PULLEYS.  117 

length  of  an  arc  of  the  pitch  line  comprising  a 
tooth  and  space.  Millwrights  ordinarily  measure 
the  pitch  as  a  cord  of  this  arc,  and,  except  in 
pinions  with  very  few  teeth,  the  two  measure- 
ments sensibly  coincide. 

Having  the  diameter  of  a  wheel,  and  the  num- 
ber of  teeth,  the  pitch  may  be  found,  as  follows  : 

Let  D  be  the  diameter  of  a  wheel,  N  the  number 
of  teeth,  and^>  the  pitch  ;  then,  as  3-1416  D  =  the 
circumference  of  the  circle, 

_  8*1416  n 

or  approximately, 

_22j> 
:  7N 

Conversely,  if  the  pitch  of  a  wheel  be  given,  and 
the  number  of  teeth,  then  the  diameter  may  be 
found, 

p  N         7  N  p 


3-1416          22 
And   if  the  pitch  and  diameter  of  a  wheel  be 
given,  then  the  number  of  teeth  may  be  found, 

3-1416  D       22  D 
N  =  -  =  -=  —  nearly. 

P  *P 

But  since  a  wheel  must  contain  a  whole  number 
of  teeth,  N  may  never  be  a  mixed  number.  If, 
therefore,  this  equation  gives  N  with  a  fraction,  a 
wheel  cannot  be  constructed  of  that  diameter  and 
pitch.  In  this  case,  however,  by  slightly  increas- 


118 


MACHINERY   OF   TRANSMISSION. 


ing  or  decreasing  either  the  diameter  or  the  pitch, 
the  necessary  conditions  may  be  complied  with. 

In  practice  it  is  convenient  to  limit  the  number 
of  pitches,  with  a  view  to  the  reduction  of  the 
number  of  patterns  required  for  casting.  Thus, 
the  following  series  gives  all  the  most  ordinary 
pitches  of  my  own  practice  : — 
Spur  flywheels,  5,  4|,  4,  3J,  3£,  3,  2 J,  2, 1£  inches. 
Spur  and  bevel  wheels,  5,  4J,  4,  3J,  3J,  3,  2f,  2J, 

2i  2|,  2,  If,  If,  If,  If,  1J,  H,  1,  I  inches. 
Wheels  of  smaller  pitch  than  this  are  not  used  in 
mill-work ;  but  in  machines,   &c.,  the  following 
pitches  would  probably  be  sufficient,  viz  : 

li  I-  f ,  i>  i  i,  iQch. 

The  value  of  *  =  ~^  ordinarily  employed  is  not 
very  accurate ;  hence  it  is  convenient  to  calculate 

P 


beforehand  the  values  of 


,31416 
and for 


31416  p 

the  most  useful  pitches. 

The  following  table  gives  these  values : 


Pitch  in 

3-1416 

Pitch 

Pitch  in 

31416 

Pitch 

inches. 

Pitch. 

31416. 

inches. 

Pitch. 

3-1416. 

5 

0-6283 

1-5915 

1% 

1  7952 

0-5570 

4>i 

0-6981 

1-4270 

1% 

1-9264 

0-5141 

4 

0-7864 

1-2732 

IK 

2-0944 

0-4774 

3>i 

0-8916 

1-1141 

1% 

2-2848 

0-4377 

3* 

0-9666 

1  0345 

IK 

2-5132 

0-3978 

3 

1  0472 

0-9548 

IH 

2-7924 

0-3580 

2* 

1  1333 

0-8754 

i 

3-1416 

0-3182 

2* 

1-2566 

0-7958 

% 

3-5904 

0-2785 

2* 

1-3963 

0-7136 

X 

4-1888 

0-2386 

2 

1-6T08 

0-6366 

5-0265 

0-1988 

1% 

1-6755 

0-5937 

~ 

6-2832 

0-1691 

WHEELS  AND  PULLEYS.  119 

EULE  1. — Given  the  pitch  and  number  of  teeth 
in  a  wheel  to  find  its  diameter. 

Multiply  the  number  of  teeth  by  the  constant 
in  the  third  or  sixth  column  of  the  preceding 
table  corresponding  to  the  pitch. 

EULE  2. — Given  the  pitch  and  diameter  of  a 
wheel  to  find  the  number  of  teeth. 

Multiply  the  diameter  by  the  constant  in  the 
second  or  fifth  column  of  the  table  corresponding 
to  the  pitch. 

If  this  rule  gives  a  mixed  number,  or  whole 
number  and  fraction,  a  wheel  cannot  be  con- 
structed, as  before  said.  The  most  convenient 
way  of  proceeding  in  that  case  will  be  to  take  the 
nearest  whole  number  to  the  number  given  by 
the  rule,  and,  using  Eule  1,  find  a  new  diameter 
which  will  differ  but  slightly  from  the  one  pre- 
viously assumed.  This  new  diameter  must  be 
taken  for  the  pitch  circle  in  constructing  the 
wheel. 

Thus,  suppose  it  required  to  find  the  diameter 
of  a  wheel  of  2  inches  pitch  and  150  teeth.  By 
Eule  1,  we  have  D  =  150  x  0-6366  =  95£  inches 
=  7  ft.  11£  inches. 

Or,  required  the  number  of  teeth  in  a  wheel  of 
3  inches  pitch  and  9  feet  diameter.  By  Eule  2 
N=  108  x  1-0472  =  113-097.  Here  the  wheel 
will  contain  very  nearly  113  teeth ;  but  if  we  wish 
to  know  more  accurately  the  diameter  of  a  wheel 
of  3  inches  pitch  and  113  teeth,  we  find  by  the 


120  MACHINERY  OF   TRANSMISSION. 

1st  Eule,  D  =  113  x  0-9548  =  107'89  inches  =  8 
feet  11/0  inches.  That  is,  a  wheel  of  exactly  9 
feet  could  not  be  constructed  with  a  3-inch  pitch, 
but  one  of  8  feet  11T90  inches  might  and  would 
contain  113  teeth. 

Professor  Willis  has  employed  another  method 
of  graduating  the  sizes  of  wheels.  Suppose  the 
diameter,  instead  of  the  circumference,  to  be 
divided  into  as  many  equal  parts  as  the  wheel  has 
teeth,  and  let  one  of  these  parts  be  called  the 
diametral  pitch  of  the  wheel,  to  distinguish  it 
from  the  common  or  circular  pitch.  Let  M  be 
the  diametral  pitch,  so  that 
D 

5  =  M 

and  let  a  series  of  values  be  taken  for  M  in  simple 
fractions  of  an  inch,  so  that 

1 

M=  — 

m 

where  N  and  m  are  always  whole  numbers. 

The  ordinary  values  of  m  are  20, 16, 14, 12, 10, 9, 
8,  7,  6,  5,  4,  3,  2,  1,  which  include  wheels  in  which 
the  circular  or  common  pitch  varies  from  £  inch 
to  3  inches,  as  shown  in  the  following  table,  given 
by  Professor  Willis : 


WHEELS   AND   PULLEYS. 


121 


Circular 

Circular 

Circular 

Circular 

Value 

Pitch  in 

Pitch  to 

Value 

Pitch  in 

Pitch  to 

of  TO. 

inches  and 

nearest 

of  m. 

inches  and 

nearest 

decimals. 

one  sixteenth 

decimals. 

one  sixteenth 

3 

1-047 

1 

9 

•349 

4 

•785 

t 

10 

•314 

A 

5 

•628 

1 

12 

•262 

6 

•524 

i 

14 

•224 

— 

7 

•449 

A 

16 

•196 

A 

8 

•393 

1 

20 

•157 

£ 

This  system  is  convenient  where  wheels  of  small 
pitch  are  employed,  and  involves  less  calculation 
than  the  common  system. 


D 


a-  ^ 

Since    -  =M,  we  have  M  = 


P 


There- 


3-1416 ' 

fore,  in  the  previous  table  (p.  118)  the  quantities  in 
the  third  and  sixth  columns  are  the  diametral 
pitches  corresponding  to  the  circular  pitches  in 
the  first  column,  and  the  numbers  in  the  second 
column  are  the  corresponding  values  of  m.  In 
fact,  this  scheme  differs  from  the  first  simply  by 
expressing  in  small  whole  numbers  the  quantity 

3-1416  . 

-  instead  of/?. 
P 
The  following  table  (pages  122  and  123)  gives 

the  relation  of  diameter,  pitch,  and  number  of 
teeth,  for  wheels  of  from  J  inch  to  five  inches 
pitch,  and  of  from  12  to  200  teeth.  Intermediate 
numbers  may  be  found  by  direct  proportion,  by 
multiplying  the  number  given  for  a  wheel  of  half 


122 


MACHINERY   OF   TUANSMISSION. 


£ 

K 
O 

s 
c 

t 
f~ 


o 
H 
cc 


WHEELS   AND   PULLEYS.  123 


124  MACHINERY   OF  TRANSMISSION". 

or  a  third  of  the  number  of  teeth  by  two  or  three, 
or  by  adding  together  the  diameters  given  for 
two  wheels  the  sum  of  whose  teeth  is  the  number 
required.  For  an  odd  number  of  teeth,  add  the 
number  given  at  the  head  of  the  table  as  many 
times  as  may  be  necessary  to  the  diameter  for  3 
wheel  of  the  nearest  number  of  teeth  given. 

The  Principles  which  Determine  the  Proper  Form 
of  the  Teeth  of  Wheels. 

The  problem  which  presents  itself  in  the  con- 
struction of  the  teeth  of  wheels,  is  to  discover  the 
curvature  which  they  should  have  in  order  that 
they  shall  revolve  through  the  action  of  the  teeth 
in  precisely  the  same  manner  as  they  would  by 
the  rolling  of  the  circumferences  of  their  pitch 
lines. 

The  general  principle  by  which  this  uniformity 
of  motion  is  secured  is  as  follows: — When  wheels 
in  gear  act  on  each  other  so  that  a  line  perpendi- 
cular to  the  common  tangent  of  the  surfaces  of 
the  teeth  at  the  point  of  contact  passes  always 
through  the  point  where  the  pitch  circles  cut  the 
line  of  centres,  they  will  exert  mutually  the  same 
force,  move  with  uniform  velocity,  and  be  of  true 
figure. 

Or,  in  other  words,  the  teeth  will  be  right! r 
constructed  when  a  line  drawn  from  the  point  of 
contact  of  the  pitch  circles  to  the  point  of  conta  t 


WHEELS   AND   PULLEYS.  125 

of  two  teeth  is  a  normal  to  the  surfaces  in  contact 
in  all  positions  of  the  wheel  and  pinion. 

Thus,  let  fig.  93  represent  a  wheel  and  pinion 
in  gear,  and  let  B  A,  A  F  be  the  primitive  radii, 
and  therefore  A  K  L  and  A  M  N  the  pitch  lines. 
Then  if  the  teeth  touch  in  c  and  D,  and  the  lines 
A  c,  A  D  be  always  perpendicular  to  the  common 
tangent  to  the  touching  parts,  the  teeth  will  be  of 
true  figure. 

Epicycloidal  Teeth 

The  epicycloid  is  the  curve  traced  by  a  fixed 
point  in  the  circumference  of  a  circle,  which  rolls 
over  or  within  the  circumference  of  another  circle, 
or  on  a  straight  line.  Thus,  let  the  circle  ABC 
be  fixed,  and  let  the  circle  ODE  roll  over  its  cir- 
cumference, then  a  point  c  in  the  circumference 
of  this  the  generating  circle  will  describe  an  epi- 
cycloid c,  c',  c",  c'",  c"",  without  the  circle  ABC. 
Similarly,  a  point  F  on  the  circumference  of  a  gen- 
erating circle  F  G,  rolling  within  the  circum- 
ference of  A  B  c,  will  describe  an  interior  epicy- 
cloid or  hypocycloid  F,  F',  F",  F'". 

The  remarkable  properties  of  the  epicycloid 
which  determine  its  fitness  for  describing  the  teeth 
of  wheels  are :  1st,  when  the  generating  circle  is 
half  the  diameter  of  the  base  circle,  and  rolls 
within  it,  the  hypocycloid  is  a  straight  line  form- 
ing a  diameter  of  the  base ;  2nd,  if  through  the 
points  of  contact  of  the  generating  circle  and  the 
11* 


126  MACHINERY  OF   TRANSMISSION". 

base,  and  the  point  describing  the  epicycloid, 
straight  lines  be  drawn,  these  straight  lines  will 
be  perpendicular  to  the  curvature  of  the  epicycloid 
from  the  point  of  contact  B  to  the  describing  point 
at  these  points.  Thus,  for  example,  B  c'"  arawn 

Fig.  94. 


Cr",  is  a  normal  to  the  curve  at  that  point ;  and 
similarly  A  F'  is  a  normal  to  the  curve  at  F'. 

Suppose  in  the  same  plane  three  circles  R  X  Y 
(fig.  95),  which  touch  each  other  in  the  point  A, 
and  whose  centres  F  B  G  are  consequently  in  a 


WHEELS   AND    PULLEYS.  127 

straight  line.  Let  one  of  these  circles  be  made  to 
revolve  round  its  centre,  and  force  the  other  two 
to  turn  round  their  centres,  which  we  suppose  to 


Fig.  96. 


be  fixed,  moving  these  circles  by  the  point  of 
continual  contact  A,  common  to  the  three  circum- 
ferences; it  is  evident  that  all  the  parts  of  the 
circumference  of  the  circle  made  to  revolve  will 
be  applied  in  succession  to  every  part  of  the  cir- 
cumferences of  the  other  two  circles,  in  the  same 
manner  as  if  the  two  circles  R  and  x  remained 
immovable,  while  the  third,  Y,  revolved  on  the 
circumferences  of  the  other  two.  Hence,  if  we 
suppose  a  style  fixed  to  the  circumference  of  the 


128  MACHINERY   OF   TRANSMISSION. 

circle  Y,  movable  round  its  centre,  the  three  circles 
having  been  obliged  to  turn  by  the  motion  of  the 
one  which  has  carried  along  the  other  two ;  when 
the  style  is  at  E,  each  of  the  two  arcs  A  c  and  A  H 
be  made  equal  to  the  arc  A  E,  the  style  will  have 
described  on  the  movable  plane  of  the  circle  R,  on 
the  exterior  part  of  which  it  revolves  a  portion 
c  E  of  an  epicycloid,  and  on  the  movable  plane  of 
the  circle  X,  within  which  we  may  consider  it  to 
revolve,  a  portion  E  H  of  a  hypocycloid.  (Camus.) 

These  two  epicycloids  traced  out  at  the  same 
time  by  the  style  E  affixed  to  the  circle  Y,  will 
touch  each  other  in  the  point  E ;  for  the  straight 
line  A  E  drawn  through  A,  where  the  generating 
circle  Y  touches  its  bases  R  and  X,  will  be  a  normal 
to  the  two  epicycloids.  The  same  will  be  true  in 
every  position  of  the  circles,  viz. :  that  the  epicy- 
cloid and  hypocycloid  will  have  a  common  normal 
passing  through  A.  Hence,  if  E  c  and  E  H  be  the 
faces  of  two  teeth  on  the  wheel  and  pinion  R  and 
X  respectively,  the  condition  of  uniform  motion 
already  given  will  be  complied  with,  the  teeth  will 
be  of  true  form,  and  if  the  hypocycloid  E  H  be 
moved  by  the  epicycloid  E  c,  or  vice  versa,  the 
wheel  and  pinion  R  and  X  will  move  precisely  as 
if  they  rolled  together  at  their  pitch  circles. 

Wheels  usually  have  their  teeth  constructed  of 
such  a  form,  that  the  flanks  or  parts  within  the 
pitch  circle  are  bounded  by  straight  lines  radii  of 
the  pitch  circles.  Bearing  in  mind  the  property 


WHEELS   AND   PULLEYS.  129 

already  stated,  that  the  hypocycloid  described  by 
a  generating  circle  of  half  the  diameter  df  the  base 
is  a  straight  line  forming  a  diameter  of  the  base, 
we  may  so  arrange  our  generating  circle  in  de- 
scribing the  teeth  of  wheels  as  to  comply  with  the 
above  rule.  By  taking  a  generating  circle  Y  of 
diameter  equal  to  the  radius  of  the  base  x,  the 
hypocycloid  E  H  will  be  part  of  a  radius  of  x ;  or, 
in  other  words,  a  radius  B  H  of  x  will  always 
touch  the  epicycloid  c  E  described  without  the 
circle  R,  by  a  generating  circle  Y,  of  a  diameter 
equal  to  the  radius  of  x.  And  the  angle  B  E  A 
being  the  angle  of  a  semicircle,  will  always  be  a 
right  angle.  That  is,  the  perpendicular  to  the 
straight  line  B  H,  at  the  point  of  contact  with 
the  epicycloid  E  c,  will  always  pass  through  A. 

We  have  hitherto  supposed  the  circles  moved 
by  contact  at  the  point  A,  in  order  to  explain  the 
generation  of  the  epicycloid  c  E  and  straight  line 
E  H  ;  but  if  we  suppose  these  already  described, 
the  former  being  fixed  to  the  circle  R,  and  the 
latter  to  the  circle  X ;  then  if  E  H  roll  by  contact 
on  the  epicycloid  c  E,  it  will  move  the  circle  R 
precisely  in  the  same  manner  as  if  the  circle 
were  moved  by  contact  at  A. 

Construction  of  Epicycloidal  Teeth. 
Since   every  tooth  in  a  wheel  is  of  precisely 
the  same  form,  it  is  sufficient  to  construct  a  sin- 
gle pattern  tooth  of  true  epicycloidal  curvature, 


130  MACHINERY   OF   TRANSMISSION. 

which  may  be  used  in  setting  out  all  the  other 
teeth. 

Fig.  96. 


First  method,  when  the  generating  circle  is  the 
same  for  wheel  and  pinion,  the  face  of  the  tooth 
an  epicycloid,  and  the  flank  a  hypocycloid. 

Construct  two  templets  A  and  B  (figs.  96,  97) 
having  their  faces  arcs  of  the  pitch  circle  of  the 
wheel  for  which  the  tooth  is  required,  and  a  third 
templet  c  cat  to  an  arc  of  the  intended  generating 


WHEELS   AND   PULLEYS. 


131 


circle  of  the  epicycloid.  Fix  a  steel  tracing  point 
p  in  the  edge  of  the  templet  c,  and  for  conveni- 
ence a  board  F  on  which  to  draw  the  tooth,  may 
be  fixed  beneath  the  templet  B.  Mark  off  on  the 
board  F  (fig.  96)  the  pitch  circle  of  the  wheel  D  E, 
and  take  distances  a  b,  b  c  equal  to  the  pitch  of 
the  teeth,  and  distances  a  a',  b  V  equal  to  the 
thickness  of  the  teeth.  If  then  the  templet  c  be 

Fig.  97. 


placed  touching  B,  and  with  the  tracing  point  p 
coinciding  with  one  of  the  marks  as  «,  and  be 
rolled  toward  E,  the  point  will  trace  out  an  epicy- 


132  MACHINERY   OF   TRANSMISSION. 

cloid  a  p  on  the  board  F,  which  will  form  one 
face  of  the  tooth.  Next  let  the  point  p  be  made 
to  coincide  of,  and  the  templet  c  be  rolled  toward 
D,  the  other  face  of  the  tooth  will  be  described. 

To  draw  the  flanks,  the  templet  A  must  now  be 
fixed  on  the  board  F,  with  its  face  in  contact  with 
B ;  remove  B  and  describe  hypocycloids  (fig.  97) 
from  a  and  a',  by  rolling  c  on  the  inside  of  the 
pitch  circle. 

The  length  of  the  teeth  ia  usually  fixed  as  a 
proportional  part  of  the  pitch,  but  the  least  neces- 
sary length  may  be  found  experimentally  by 
replacing  the  templet  B  on  the  board  F,  and 
making  p  coincide  with  a,  roll  C  toward  E  till  it 
touches  B  in  b,  the  corresponding  face  of  the  next 
tooth ;  mark  then  the  position  of  the  tracing  point 
and  through  this  point  draw  an  arc  from  the 
centre  g  of  the  wheel :  this  arc  will  mark  the  ex- 
tremity of  the  tooth,  and  the  arc  g  p  will  be  the 
true  radius  of  the  wheel. 

This  process,  which,  though  complicated  in 
description,  is  very  easy  in  practice,  must  be 
repeated  with  two  templets  cut  to  the  pitch  circle 
of  the  pinion,  the  same  generating  circle  c  being 
employed ;  a  similar  pattern  tooth  will  thus  be 
found  for  the  pinion,  which  will  work  with  that 
already  found  for  the  wheel.  The  usual  custom 
in  practice  is  for  the  millwright  first  to  describe 
the  epicycloidal  and  hypocycloidal  forms  of  the 
teeth  required  in  the  wheel  and  pinion ;  he  then 


WHEELS  AND  PULLEYS.  133 

constructs  two  model  teeth,  one  for  the  wheel  arid 
the  other  for  the  pinion,  and  from  these  he  deter- 
mines the  true  curves,  and  by  means  of  his  com- 
pastes  transfers  the  same  to  the  wheels  or  patterns 
on  which  these  forms  are  to  be  impressed.  The 
generating  circle,  it  may  be  observed,  must  not 
exceed  in  size  the  radius  of  the  pinion,  or  it 
would  give  rise  to  a  weak  form  of  tooth,  thinner 
at  the  root  than  at  the  pitch  circle. 

Second  method,  where  two  generating  circles  are 
employed,  in  order  that  the  flanks  of  the  teeth 
may  be  straight  lines  radii  of  the  wheel  and 
pinion  respectively. 

It  is  the  usual  practice  of  millwrights  to  make 
the  parts  of  the  teeth  of  wheels  within  the  pitch 
circles  radii  of  the  wheel.  Now,  we  have  seen 
that  a  hypocycloid  described  by  a  generating 
circle  equal  in  diameter  to  the  radius  of  the 
wheel  would  be  a  diameter  of  the  wheel.  If, 
therefore,  the  flank  of  the  tooth  of  the  wheel  and 
the  face  of  the  tooth  of  the  pinion  be  described 
by  a  templet  cut  to  a  radius  equal  to  half  that  of 
the  wheel  and  the  flank  of  the  tooth  of  the  pinion 
and  face  of  that  of  the  wheel  be  described  by  a 
templet  cut  to  a  radius  equal  to  half  that  of  the 
pinion,  then  these  teeth  will  work  together  truly, 
and  will  have  radial  flanks. 

|     Since  it  is  unnecessary  to  describe  the  flanks  of 
such  teeth  by  templets,  there  will  be  needed  only 
one  templet  cut  to  the  pitch  circle  of  each  wheel, 
12 


MACHINERY   OF   TRANSMISSION. 

but  templets  of  two  generating  circles  are  re- 
quired. In  other  respects  the  method  is  identical 
with  that  already  described.  The  great  defect  of 
this  method  is,  that  neither  the  wheel  nor  pinion 
will  work  accurately  with  a  wheel  or  pinion  of 
any  other  diameter  than  that  for  which  they  were 
originally  made,  and  thus  a  vast  number  of  wheel 
patterns  must  be  made  to  fulfil  the  requirements 
of  practice ;  whereas  wheels  described  by  the 
previous  method  will  work  equally  well  with  all 
other  wheels  the  teeth  of  which  have  been  de- 
scribed by  the  same  generating  circle — it  being 
understood  that  only  the  parts  of  teeth  without  the 
pitch  circle  of  the  wheel  roll  on  the  parts  within 
the  pitch  circle  of  the  pinion,  and  those  without 
the  pitch  circle  of  the  pinion  on  those  within  the 
pitch  circle  of  the  wheel. 

Hence  Professor  Willis  has  been  led  to  suggest 
that  for  a  given  set  of  wheels  a  constant  generat- 
ing circle  should  be  taken  to  describe  both  the 
parts  without  and  within  the  pitch  circles  of  the 
whole  series,  instead  of  making  that  circle  depend 
on  the  diameters  of  the  wheels.  In  this  case  the 
first  solution  must  be  employed,  and  the  flanks 
of  the  teeth  will  not  be  straight;  but  the  great 
advantage  is  gained,  that  any  pair  of  wheels  in 
the  series  will  work  together  equally  well. 

To  determine  the  proper  size  of  the  generating 
circle,  we  must  remember  that  a  tooth  of  weak 
form  is  produced  when  the  generating  circle  is 


WHEELS   AND   PULLEYS.  135 

greater  than  half  the  diameter  of  the  wheel. 
Hence  the  generating  circle  may  be  best  made  of 
a  diameter  equal  to  the  radius  of  the  smallest 
pinion  of  the  series  which  are  to  work  together. 

The  Rack  is  the  extreme  case  of  a  wheel,  or 
may  be  considered  as  a  wheel  of  infinite  radius. 
It  may  be  described  by  either  of  the  methods 
above,  only  noting  that,  if  the  second  method  be 
employed,  the  generating  circle  which  traces  the 
face  of  the  teeth  of  the  wheel  becomes  a  straight 
line,  and  the  epicycloid  becomes  an  involute. 

If  the  teeth  of  a  series  of  wheels  and  of  a  rack 
be  described  by  the  same  generating  circle,  any 
of  the  wheels  will  work  with  equal  accuracy  into 
the  rack. 

Involute  Teeth. 

The  Involute. — The  curve  traced  by  a  flexible 
line  unwinding  from  the  circumference  t>f  a  circle, 
is  called  an  involute. 

Let  P  and  w  (fig.  98)  be  the  pitch  lines  of  a 
wheel  and  pinion,  and  let  A  and  B  be  their  centres. 
From  A  and  B  describe  two  circles  D  c,  with  radii 
A  b  and  B  b  of  the  wheel  and  pinion  respectively  ; 
so  that 

Ac  :  Be  ::  A  D  :  BC 

Let  m  n  and  op  be  two  involute  curves  described 
by  flexible  lines  unrolling  from  the  circles  D  and 
c  respectively,  and  touching  at  b.  Then  if  b  c,  b  D 
be  drawn  tangents  to  the  circles  at  the  points  D 


136  MACHINERY   OF  TRANSMISSION. 

and  c,  they  are  also  in  one  straight  line,  because 
they  are  both  normals  to  the  curves  at  b.  It  may 
also  be  shown  that  the  line  c  D  intersects  A  B  in  c, 
where  the  pituh  lines  touch.  Hence  we  have 
found  two  curves  such,  that  the  line  perpendi- 
ng. 98. 


cular  to  their  common  tangent  passes  in  all  posi- 
tions of  the  wheel  and  pinion  through  c,  which  is 
the  sufficient  condition  of  their  uniform  motion, 
if  moved  by  the  sliding  of  the  curves  instead  of 
by  contact  at  c.  Hence,  if  the  wheels  be  con- 


WHEELS   AND   PULLEYS.  137 

structed  with  teeth  formed  to  these  involute 
curves,  they  will  work  with  perfect  regularity  of 
motion. 

In  practice,  the  chief  condition  to  be  observed 
is  to  diminish  the  pressure  on  the  axes,  which  is 
the  chief  defect  of  this  form  of  teeth.  The  com 
mon  tangent  should  be  drawn  through  c,  making 
an  angle  with  A  B,  not  deviating  more  than  20° 
from  a  right  angle.  Involute  wheels  have  the 
double  advantage  that  they  work  equally  well  if, 
through  the  wear  of'  the  brasses,  the  wheels  have 
receded  from  one  another;  and  any  involute 
wheels  of  the  same  pitch  and  similarly  described 
— that  is,  having  the  common  tangent  to  the  base 
circles  passing  through  the  point  of  contact  of 
the  pitch  lines ;  or,  in  other  words,  base  circles 
proportional  to  the  primitive  radii — will  woik 
together. 

Mr.  Hawkins,  the  translator  of  Camus,  first 
proposed  a  simple  instrument  for  describing  the 
teeth  of  wheels  to  an  involute  curve.  It  consists 
of  a  straight  piece  of  watch-spring  a  b  (fig.  99), 
with  a  screw  at  one  end,  and  filed  away  at  the 
edges  so  as  to  leave  two  teeth  or  tracers,  c  c,  pro- 
jecting from  the  edges  of  the  watch-spring.  At 
b  a  bit  of  wire  is  put  through,  and  riveted,  so  as 
to  form  a  knot  by  which  the  spring  can  be  firmly 
held  and  stretched,  as  it  is  unwound  from  the  base 
on  which  the  involute  is  generated.  This  watch- 
spring  is  screwed  to  the  edge  of  a  templet  A, 


138  MACHINERY    OF   TRANSMISSION. 

curved  to  the  radius  of  the  base  circle  of  the  in- 
volute; and  this  being  placed  so  that  its  centre 
coincides  with  the  centre  of  the  wheel,  and  re- 
volved to  bring  one  of  the  tracing  points  c  in 
succession  to  each  of  the  points  at  which  corres- 
ponding faces  of  the  teeth  cut  the  pitch  line,  a 

Fig.  99. 


series  of  involute  curves  may  be  described  by 
unfolding  the  watch-spring,  whilst  keeping  it 
firmly  stretched  tangentially  to  the  sector  to  which 
it  is  fixed.  The  sector  A  must  then  be  turned 
over,  and  the  involutes  of  the  opposite  faces  of 
the  teeth  struck  in  a  similar  manner. 

Another  plan  is  to  employ  a  straight  ruler  in- 
stead of  the  watch-spring,  a  tracer  being  fixed  in 
its  edge.  This  shows  that  the  involute  is  an  epi- 
cycloid generated  by  a  straight  line.  The  ruler 
must  be  kept  in  contact  with  the  base  circle,  and 


WHEELS   AND   PULLEYS.  139 

the  tracer  brought  in  succession  to  all  the  points 
in  which  the  faces  of  the  teeth  cut  the  pitch  line. 

Hence,  to  describe  a  wheel  with  involute  teeth, 
the  line  of  centres  must  be  be  drawn  and  divided 
proportionally  to  the  number  of  teeth  in  the  wheel 
and  pinion.  Draw  the  pitch  line;  divide  the 
pitch  line  into  the  same  number  of  equal  parts  as 
there  are  teeth  in  the  wheel,  and  at  these  points 
mark  out  the  thicknesses  of  the  teeth  all  round. 
Draw  the  tangent  to  the  base  circles,  making  an 
angle  of  about  80°  with  the  line  of  centres,  which 
will  give  the  radius  of  the  base  circle-  drawn 
touching  it.  A  templet  must  be  made  to  this 
radius,  and  then  the  involutes  may  be  drawn  by 
either  of  the  preceding  methods. 

Allowance  must  be  made  to  permit  free  play  of 
the  teeth  in  the  spaces,  the  teeth  being  somewhat 
shorter  than  the  distance  between  the  bases  of  the 
involutes.  But  wheels  of  this  figure  require  but 
little  play  in  the  engagement. 

In  the  case  of  racks,  the  rack-teeth  are  bounded 
by  straight  lines  perpendicular  to  the  tangent 
drawn  from  the  point  where  the  pitch  lines  touch, 
to  the  base  circle  from  which  the  involutes  of  the 
wheel  are  struck.  If  the  teeth  of  the  rack  be 
made  rectangular — that  is,  bounded  by  lines  per- 
pendicular to  the  pitch  line— the  involute  must  be 
struck  from  a  base  circle  equal  to  the  pitch  circle 
of  the  wheel.  In  the  former  case  there  is  a  down- 
ward pressure  on  the  rack ;  in  the  latter,  the  teeth 


140  MACHINERY   OF  TRANSMISSION.. 

of  the  wheel  touch  those  of  the  rack  in  a  single 
point — namely,  the  pitch  line  of  the  latter. 

Professor  Willis's  Method  of  Striking  the  Teeth  of 
Wheels. 

In  practice,  the  custom  of  describing  the  teeth 
of  wheels  as  arcs  of  circles,  has,  from  its  simplic- 
ity, been  generally  adopted.  The  methods  already 
given,  however  simple,  when  adopted  in  the  form- 
ation of  a  single  tooth,  become  tedious  in  their 
application  to  wheels  of  large  size;  and  to  this 
must  be  added  the  imperfect  comprehension  of 
their  advantages  by  the  millwrights  charged  with 
the  task  of  designing  wheel  patterns. 

Circular  arcs  struck  at  random,  according  to  the 
judgment  of  the  millwright,  are  often  employed; 
and  even  where  better  principles  have  been  intro- 
duced, it  is  common,  after  describing  a  single 
tooth  accurately,  to  find  by  trial  a  circular  arc 
nearly  corresponding  with  its  curve,  and  to  em- 
ploy this  in  marking  out  the  cogs  of  the  required 
wheel. 

Seeing  the  advantages  of  the  circular  arc,  and 
believing  that  it  is  not  objectionable  if  only  the 
employment  of  it  is  guided  by  true  principles, 
Professor  Willis  has  rendered  this  great  service 
to  practical  mechanics — he  has  shown  how,  by  a 
simple  construction,  the  arcs  of  circles  may  be 
found,  which,  used  in  the  construction  of  the  teeth 
of  wheels,  will  work  truly  on  each  other. 


WHEELS   AND   PULLEYS.  141 

Let  A  B  (fig.  100)  be  the  centres  of  a  wheel  and 
pinion,  and  c  the  point  of  contact  of  the  pitch 
circles  on  the  line  of  centres.  Through  c  draw 
c  Cc  at  any  angle  with  .A  B.  Assume  c  as  the 
centre  from  which  to  describe  an  arc  for  a  tooth 
of  the  wheel  a.  Draw  c  D  perpendicular  to  c  c  c', 
and  from  A  through  c  draw  A  c  D,  meeting  c  D  in 


Fig.  100. 


D.  Lastly,  from  D  through  B  draw  D  B  c,  meeting 
c  c  c'  in  c.  Then  a  small  arc  drawn  from  c  with 
radius  c  c  as  a  tooth  for  the  wheel  a,  will  work 
correctly  with  a  small  arc  drawn  from  c',  with  a 
radius  c'  c  as  a  tooth  for  the  wheel  B.* 

Professor  Willis  recommends  75°  30'  as  the 
best  magnitude  of  the  angle  ACC,  so  that  Cos.  75° 
30'  =  J.  If  this  angle  be  constant  in  a  set  of 
wheels,  any  two  will  work  truly  together. 

*  Willis's  "Principles  of  Mechanism,"  p.  123. 


MACHINERY   OF   TRANSMISSION. 
Fig.  101. 


Tables  showing  the  place  of  the  Centres 

upon  the  Scales 

Centres  for  the  Flanks  of  Teeth 

Number 
f.e 

Pitch  in  inches 

or 

Teeth 

1 

M 

I* 

If 

2 

2| 

2* 

3 

13 

129 

160 

193 

225 

257 

289 

321 

386 

14 

69 

87 

104 

121 

139 

156 

173 

208 

15 

49 

62 

74 

86 

99 

111 

123 

148 

16 

40 

50 

59 

69 

79 

89 

99 

191 

17 

34 

42 

50 

59 

67 

75 

84 

101 

18 

30 

37 

45 

52 

59 

67 

74 

89 

20 

25 

31 

37 

43 

49 

56 

62 

74 

22 

22 

27 

33 

39 

43 

49 

54 

65 

24 

20 

25 

30 

35 

40 

45 

49 

59 

26 

18 

23 

27 

32 

37 

41 

46 

55 

SO 

17 

21 

25 

29 

33 

37 

41 

49 

4Q 

15 

18 

21 

25 

28 

32 

35 

42 

60 

13 

15 

19 

22 

25 

28 

31 

37 

80 

12 

e  m 

17 

20 

23 

26 

29 

35 

100 

11 

14 

22 

25 

28 

34 

150 

13 

16 

i<j 

21 

24 

27 

32 

Hack 

10 

12 

15 

17 

20 

22 

25 

30 

Centres  for  the  Faces  of  Teeth 

12 

5 

6 

7 

9 

10 

11 

12 

15 

15 

i  . 

7 

8 

10 

11 

12 

14 

17 

20 

6 

8 

9 

11 

12 

14 

15 

18 

30 

7 

9 

10 

12 

14 

16 

18 

21 

40 

8 

•  * 

11 

13 

15 

17 

19 

23 

60 

9  9 

10 

12 

14 

16 

18 

20 

25 

80 

9 

11 

13 

15 

17 

19 

21 

26 

100 

.. 

18 

20 

22 

150 

ii 

ie 

19 

21 

23 

27 

Rack 

10 

12 

15 

17 

20 

22 

25 

30 

CO 
o 
P 


The  figure  is  of  half  the  linear  dimensions  of  the 
original    . 


WHEELS  AND   PULLEYS.  143 

For  the  easier  description  of  these  teeth,  Pro- 
fessor Willis  has  invented  the  odontograph,  a 
simple  instrument  of  graduated  card  or  wood,  by 
which  the  position  of  the  centres  and  radii  of  the 
arcs  of  the  teeth  can  very  easily  be  found.  This 
instrument*  is  of  the  form  shown  in  fig.  101,  of 
half  its  proper  lineal  dimensions.  It  has  the  bot- 
tom edge  bevelled  off  at  an  angle  of  75°.  The 
point  where  this  would  cut  the  right-hand  edge  is 
the  zero  of  the  scales.  These  scales  are  graduated 
to  twentieths  of  an  inch,  to  avoid  fractional  parts 
in  the  tables,  and  depart  in  each  direction  from 
the  zero,  the  upper  being  that  employed  in  find- 
ing the  centres  of  the  flanks  of  the  teeth  or  parts 
within  the  pitch  circle,  and  the  lower  for  finding 
the  centres  of  the  faces  of  the  teeth  or  parts  with- 
out the  pitch  circle.  Tables  are  given  on  the 
odontograph  for  finding  the  graduation  on  the 
scale  corresponding  to  any  given  pitch  and  num- 
ber of  teeth.  For  intermediate  pitches,  not  given 
in  the  table,  or  for  wheels  of  greater  size,  the  cor- 
responding numbers  can  be  found  by  simple  pro- 
portion. For  wheels  of  only  twelve  teeth  the 
flanks  are  straight,  and  form  parts  of  radii  of  the 
pitch  circle. 

In  fig.  102,  let  A  be  the  centre  of  a  wheel,  K  G?L 
the*  pitch  line.  Set  off  K  L  equal  to  the  pitch,  and 

*Professor  Willis's  Odoutograph  may  be  obtained  of 
Messrs.  Holtzapfel  of  London. 


144  MACHINERY   OF   TRANSMISSION. 

bisect  it  in  d.  Draw  radii  A  K,  A  L.  Place  the  odon- 
tograph  with  its  bevelled  edge  on  the  radius  A  K, 
and  zero  of  the  scale  on  the  pitch  line.  Then  look 
out  in  the  table  of  centres  for  the  flanks  of  teeth, 
the  number  corresponding  to  the  pitch,  and  re- 
quired number  of  teeth,  and  mark  off  this  point  h, 

Kg.  102. 


from  the  scale  of  centres  for  the  flanks  of  teeth. 
Then  remove  the  odontograph,  and  similarly  place 
it  on  the  radius  A  L.  Find  in  the  table  of  centres 
for  the  faces  of  the  teeth  the  number  correspond- 
•  ing  to  the  pitch  and  number  of  teeth  in  the  wheel, 
and  mark  it  off  at/,  on  the  scale  for  centres  of  the 
faces  of  teeth.  Then  describe  two  arcs  from  h 


ON  THE  TEETH   OF   WHEELS.  145 

and  /  with  h  d  and  /  d  as  radii ;  these  will  form 
the  side  of  a  tooth.  Then,  from  d  let  the  pitch 
line  be  marked  off  into  as  many  equal  spaces  as 
there  are  teeth  in  the  wheel,  and  these  be  divided 
proportionally  to  the  widths  of  the  teeth  and 
spaces.  Through  h  and/  with  radii  A  h  and  A/ 
draw  circles.  Take  h  d  as  a  radius,  and,  placing 
one  foot  of  the  compass  on  the  divisions  of  the 
pitch  line,  and  the  other  in  the  circle  drawn 
through  h,  describe  a  series  of  arcs  forming  the 
flanks  of  the  teeth.  Similarly  with  radius  /  d, 
and  one  leg  of  the  compass  on  the  circle  drawn 
through/,  describe  the  faces  of  the  teeth. 

For  an  annular  wheel  the  same  rules  apply, 
only  that  the  part  of  the  curve  which  is  face  in  a 
spur  wheel  becomes  the  flank  in  an  annular  wheel, 
and  vice  versa.  For  a  rack,  the  pitch  line  is 
straight,  and  A  K,  A  L  are  parallel  and  perpen- 
dicular to  it,  at  a  distance  equal  to  the  pitch. 

As  these  odontographs  may  be  purchased  in  a 
very  convenient  form,  with  tables  for  their  use, 
and  also  with  tables  of  the  widths  of  teeth,  and 
spaces  and  length  of  teeth  within  and  without  the 
pitch  circle,  it  is  not  necessary  to  describe  them 
in  further  detail  here. 

Qeneral  Form  and  Proportions  of  Teeth  of  Wheels. 

The  following  have  been  drawn  as  a  series  of 
wheels  and  racks  to  illustrate  the  general  form  of 

the  teeth  of  wheels.     The  pitch  in  figs.  103,  104, 
13 


146  MACHINERY   OF   TRANSMISSION. 

105,  and  106  is  one  inch,  and  that  in  fig.  107  is 
2|  inches. 

In  figs.  103, 104,  105,  and  106  the  wheel  is  191 
inches  diameter ;  in  fig.  107  it  is  13  feet  diameter. 

Fig.  103  represents  the  form  of  the  teeth  on 
Professor  Willis's  system,  the  curves  being  arcs  of 
circles.  Fig.  104  gives  the  form  of  epicycloidal 
teeth,  struck  by  a  single  generating  circle  rolled 
without  the  pitch  circle  for  the  faces,  and  within 
it  for  the  flanks.  This  is  the  best  system,  as  any 
pair  of  wheels  so  struck,  with  the  same  generating 
circle,  and  of  equal  pitch,  will  work  together.  Fig. 
105  shows  the  common  form  of  epicycloidal  teeth, 
the  flanks  being  straight.  In  this  case  the  faces  of 
the  rack  are  struck  by  a  generating  circle  half 
the  diameter  of  the  wheel,  and  the  faces  of  the 
wheel,  being  obtained  by  a  generating  circle  of 
infinite  diameter  or  straight  line,  become  invo- 
lutes. Fig.  106  gives  the  form  of  teeth  described 
as  involutes,  the  curve  being  continuous,  and,  in 
the  case  of  the  rack,  a  straight  line  perpendicular 
to  the  tangent  to  the  base  circle.  In  these  teeth 
it  is  possible  to  work  with  very  little  play.  They 
are  a  good  form  for  wheel  and  rack  working  to- 
gether, the  pressure  on  the  journals  being,  in  this 
case,  less  objectionable.  Fig.  107  shows  the  teeth 
of  a  large  wheel,  traced  from  one  of  my  own  pat- 
terns, to  exhibit  the  form  and  proportion  which 
practice  has  shown  to  be  desirable. 
.  In  these  teeth  the  pitch  c  d  being  2|  inches,  the 


ON  THE   TEETH   OF   WHEELS.  147 

Fig.  101. 


148  MACHINERY   OF   TRANSMISSION. 

Fig.  104. 


ON   THE   TEETH  OF   WHEELS. 
Fig.  105. 


150  MACHINERY   OF   TRANSMISSION. 

Fig.  106. 


ON   THE  TEETH   OF   WHEELS. 
Fig. 107. 


151 


152  MACHINERY   OF   TRANSMISSION. 

depth  of  the  tooth  or  distance  a  b  is  yths  or  fths  of 
the  pitch.  The  proportions  of  the  parts  may  be 
given  as  follows: — 

Proportional 

Part.  Inches. 

Pitch                                    =     cd    =     1-00  =  2£ 

Depth                                  =     ab     —     0-75  =  1£ 

Working  depth                  =    a  e    —     0-70  =  1^ 

Clearance                            =     e  b     =    0'05  =  £ 

Thickness                            =     c  /    =     0-45  =  1£ 

Width  of  space                  =    fd    =    0-55  =  1§ 

Play  or/ d,  cf                                   =    0-10  =  ± 

Length  beyond  pitch  line  =    a  g    =    0'35  =  | 

Taking  these  proportions  we  may  construct  a 
scale  which  shall  give  directly  the  corresponding 
numbers  for  any  pitch.  Taking  a  vertical  line, 
and  dividing  it  into  eighths  of  an  inch,  we  get  the 
scale  of  pitches,  (fig.  108.)  Draw  lines  perpendic- 
ular to  this,  and  on  any  one  of  them  mark  off*  a 
series  of  distances  equal  to  the  clearance,  depth, 
thickness,  etc.,  of  the  teeth  corresponding  to  that 
pitch.  Through  o  and  these  points  draw  the  lines 
shown  in  the  figure ;  they  will  divide  the  lines 
corresponding  to  all  other  pitches  in  the  same 
proportion. 

It  is  usual  to  allow  a  greater  amount  of  clear- 
ance in  small  wheels  than  is  necessary  in  large 
ones.  Yery  varying  proportions  have  been  given 
by  different  millwrights,  y'oth,  j^th.  y'gth,  and  %0l}i 
of  the  pitch  having  been  used  in  different  circum- 
stances, even  with  the  best  mill-work.  In  the 
scale  (fig.  108,)  this  has  to  a  certain  extent  been 
taken  into  account ;  T'0th  of  the  pitch  is  allowed 


OX   THE   TEETH    OF    WHEELS. 
Fig.  108. 


153 


-HD  pth  beyond  Pitch  line  =  a  <r ™>>         j 

-{-Depth  within  Pitch  line  =  b  g >• 

-Hhickness  of  Tooth  =  c  f  

-MVidth  of  Space    =  d  f 

-^Working  depth  of  Tooth  =  a  e         

-Jf  Mole  de^th  of  Tooth  =  a  b  


154 


MACHINERY   OF   TRANSMISSION". 


in  smaller  wheels,  decreasing  to  y5th  in  the 
largest;  hence  the  lines  are  not  absolutely 
straight,  but  are  slightly  curved,  except  that  for 
the  whole  depth  of  the  tooth,  which  quantity  has 
been  assumed  to  vary  directly  as  the  pitch. 

Assuming  that  this  scale  represents  with  suffi 
cient  accuracy  the  proportions  \vhich  practice 
shows  to  be  best  in  average  cases,  we  may 
construct  a  table  for  the  guidance  of  the  mill- 
wright. From  this  he  must  vary  in  cases  where 

TABLES  OF  PROPORTIONS  OF  TEETH  OF  WHEELS  FOR  AVER- 
AOK  PRACTICE. 


Pitch. 

Clear- 
ance 
and 
play. 

Depth 
beyond 
pitch 
line. 

Depth 
within 
pitch 
Hue. 

Working 
depth. 

Whole 
depth. 

Thick- 
ness of 
tooth. 

Width 
of 
space. 

i 

•06 

•16 

•22 

•32 

•38 

•22 

•28 

f 

•08 

•25 

•33 

•50 

•58 

•33 

•42 

1 

•10 

•335 

•435 

•67 

•77 

•45 

•55 

M 

•12 

•42 

•54 

•84 

•96 

•56 

•69 

if 

•13 

•51 

•64 

1-02 

1-15 

•68 

•82 

if 

•14 

•60 

•74 

1-20 

1-34 

•80 

•95 

2 

•16 

•685 

•845 

1-37 

1-53 

•92 

1-08 

H 

•17 

•775 

•945 

1-55 

1-72 

1-04 

1-21 

2* 

•19 

•86 

1-05 

1-72 

1-91 

1-15 

1-35 

2} 

•20 

•95 

1-15 

1-90 

2-10 

1-27 

1-47 

3 

•22 

1-04 

1-26 

2-08 

2-30 

1-39 

1-61 

3^ 

•23 

1-13 

1-36 

2-26 

2-49 

1-51 

1-74 

3* 

•25 

1-215 

1-465 

2-43 

2-68 

1-62 

1-88 

3f 

•26 

1-305 

1-565 

2-61 

2-87 

1-74 

2-01 

4 

•28 

1-39 

1-67 

2-78 

3-06 

1-86 

2-14 

** 

•31 

1-565 

1-875 

3-13 

3-44 

2-09 

2-40 

5 

•34 

1-745 

2-085 

3-49 

3-83 

2-33 

2-67 

5* 

•37 

1-925 

2-295 

3-85 

4-21 

2-56 

2-93 

6 

•40 

2-10 

2-50 

4-20 

4-60 

2-80 

3-20 

ON   THE  TEETH    OF   WHEELS. 


155 


it  appears  necessary  to  allow  more  for  defects  of 
workmanship,  or  to  permit  less  "  backlash  ;"*  it 
being  understood  that  the  table  will  only  apply  in 
cases  where  the  teeth  are  formed  with  an  approxi- 
mation to  the  true  mathematical  figure. 

In  wood  and  iron  gear  where  the  teeth  are  care- 
fully cut,  very  little  if  any  clearance  is  necessary, 
as  they  work  much  better  when  the  tooth  of  each 
wheel  fills  their  allotted  spaces.  It  is,  however, 
different  where  wheels  have  to  gear  together  di- 
rect from  the  foundry,  where  the  teeth  are  not 
unfrequently  deranged  in  the  act  of  moulding  in 
the  sand. 

This  table  gives  the  number  to  the  nearest  hun- 
dredth of  an  inch.  It  may  be  converted  into  the 
ordinary  scale  of  eights  by  the  following  table  : — 


Thirty  Seconds  of  an  Inch. 

1 

•031 

2 

3 

4 

6 

6 

7 

8 

9 

10 

Corresponding 
Decimal. 

•062 

0-94 

•125 

•156 

•188 

•219 

•250 

•281 

•3125 

As,  unfortunately,  decimal  scales  are  not  yet 
much  used  by  millwrights,  the  following  table  has 
been  prepared,  giving  the  numbers  in  the  preced- 
ing table  in  thirty  seconds  of  an  inch,  such  changes 


*  A  technical  expression  for  reaction  on  the  back  of  the 
teeth. 


156 


MACHINERY   OF   TRANSMISSION. 


being  made  as  will  reduce  as  much  as  possible  the 
errors  of  employing  this  rough  standard.  The 
former  table  is  to  be  preferred  where  it  can  be 
used,  but  in  other  cases  the  following  one  may  be 
relied  on.  The  left-hand  figures  in  each  column 
are  inches,  the  right-hand  ones  thirty  seconds  of 
an  inch,  the  denominators  of  the  fraction  being 
omitted. 


TABLE  GIVING  THE  PROPORTIONS  OF  TEETH  OF  WHEELS 
IN  INCHES  AND  THIRTY  SECONDS  OF  AN  INCH. 


Pitch, 
inches. 

Clearance. 

Depth 
beyond  the 
pitch  line. 

Depth 
within  the 
pitch  line. 

Working 
depth. 

Whole 
depth. 

Thickness 
of  tooth. 

i 

0"2 
0    3 

0"5 
0    8 

0"  7 
Oil 

0"10 
0    16 

0"  12 
0    19 

0"7 
0  10 

1 

0    3 

0  11 

014 

0    22 

0    25 

0  14 

11 

0    4 

013 

017 

0    26 

0    30 

0  18 

to  i—  '  i—  ' 

lH»K*-l 

0    4 
0    4 
0    5 

016 
0  19 
022 

020 
023 
027 

1      0 
1      6 
1    12 

1      4 
1     10 
1    17 

021 
025 
029 

2? 

0    5 

025 

030 

1    18 

1     23 

1    1 

24 
2| 

3 

0    5 
0    6 

0    7 

028 
0  31 
1    1 

033 
037 
1    8 

1    24 
1    30 
2      2 

1     29 
2      4 
2       9 

1    5 
1    8 
112 

3£ 

0    7 

1    4 

111 

2      8 

2     15 

116 

3| 

3| 

0    8 
0    8 

1    7 
110 

1  15 

1  18 

2    14 
2    20 

2     22 

2     28 

]  20 
123 

4 

0    9 

1  12 

1  21 

2    24 

3       1 

127 

41 

010 

118 

128 

3     4 

3     14 

2    3 

5 

0  11 

124 

1  35 

3    16 

3     27 

2  10 

5* 

6 

0  11 
012 

1  30 
2    4 

141 
2  16 

3    28 
4      8 

4      7 
4    20 

2  18 
225 

ON  THE   TEETH   OF  WHEELS. 
Bevel  Wheels. 


157 


Hitherto  we  have  considered  only  that  case  of 
toothed  wheels  in  which  the  pitch  lines  are  in  one 
plane.  We  have  now  to  examine  the  modifica- 
tions which  are  necessary  when  the  axes  of  the 


Tig.  109. 


158  MACHINERY  OF   TRANSMISSION. 

wheel  and  pinion  are  inclined.  It  was  shown  in 
the  preliminary  chapter4*  that  in  this  case  motion 
might  be  transmitted  by  the  rolling  contact  of  the 
frustra  of  two  cones.  If,  therefore,  teeth  be  ap- 
plied to  these  frustra,  in  the  same  manner  as  in 
spur  gearing,  they  are  attached  to  cylindrical  sur- 
faces, bevel  gearing  will  be  formed  acting  on  the 
same  principles  of  sliding  contact  which  we  have 
already  discussed. 

Let  A  B  c,  A  c  D  (fig.  109)  be  two  cones  rolling 
in  contact ;  take  any  other  cone  A  E  c  also  rolling 
in  contact  with  A  B  c,  in  the  line  A  c.  As  these 
cones  roll  together,  the  generating  cone  A  E  c  will 
describe  an  epicycloidal  surface  p  q  r  s  on  the  out- 
side of  the  cone  A  c  D,  and  a  hypocycloidal  sur- 
face p  t  v  s  on  the  inside  of  the  cone  A  c  D. 
These  surfaces  will  touch  in  the  line  p  s,  and  will 
have  a  plane  normal  to  their  common  tangent 
passing  through  A  c.  If,  therefore,  these  surfaces 
be  attached  respectively  to  the  cones  A  B  C,  A  c  D, 
and  the  motion  of  one  cone  be  communicated  to 
the  other  through  the  sliding  contact  of  these  sur- 
faces, the  motion  will  be  uniform,  as  if  the  cones 
were  driven  by  rolling  contact  at  A  c. 

The  curves  p  t,  p  q,  lie  in  reality  on  the  surface 
of  a  sphere  of  a  radius  equal  to  A  c ;  but  in  prac- 
tice, in  bevel  wheels,  a  small  frustrum  of  a  cone, 
tangential  to  the  sphere  at  the  circumference  of 

*  See  p.  66,  §  88,  89. 


ON   THE  TEETH   OF   WHEELS.  159 

the  pitch  line,  is  substituted  for  the  spherical  seg- 
ment. Thus  draw  F  c  G  (fig.  109)  perpendicular 
to  A  C,  cutting  the  axes  of  the  cones  in  F  and  G. 
Let  these  lines  revolve  over  the  pitch  lines  of  the 
cones  and  describe  the  narrow  frustra.  Then  the 
epicycloidal  surfaces  may,  without  sensible  error, 
be  supposed  to  lie  in  these  frustra,  and  to  be  gen- 
erated there  by  the  revolution  of  a  generating 
circle  C  E. 

Imagine  the  surface  of  these  frustra  to  be  un- 
wrapped so  as  to  lie  in  one  plane,  they  will  form 
parts  of  circular  annuli.  Thus  let  A  B  c,  A  c  D 
(fig.  110),  be  two  conical  frustra;  draw  F  c  G  as 
before,  perpendicular  to  the  line  of  contact  A  c. 
From  G,  with  radii  G  H,  G  c,  and  G  K,  describe  the 
circles  K  L,  c  M,  H  N;  and  from  F,  with  radii  F  K, 
F  c,  F  H,  describe  similar  circles  K  P,  c  Q,  H  R ; 
then  the  surfaces  K  P  B  H  and  K  L  N  H  will  be  de- 
velopements  of  the  frustra  c  D,  c  B.  Let  these  be 
treated  as  spur  wheels,  and  c  Q,  c  M  being  treated 
as  the  pitch  lines,  let  teeth  be  described  by  a  de- 
scribing circle  in  the  method  already  explained 
for  epicycloidal  or  other  teeth.  If,  then,  the  plane 
on  which  these  have  been  described,  and  which 
we  suppose  of  drawing  paper  or  other  flexible 
material,  be  cut  along  the  arcs  K  P,  H  R,  K  L,  H  N, 
the  circular  annuli  may  be  wrapped  round  the 
frustra  c  B,  c  D,  and  the  forms  of  the  teeth  traced 
oft'  upon  them. 

The  axes  of  bevel  wheels  are  in  practice,  in 


MACHINERY    OF   TRANSMISSION". 


Fig.  110. 


the  great  generality  of  cases,  at  right  angles. 
Fig.  Ill  shows  such  a  pair  of  bevels,  with  the 
frustra  of  the  extremity  of  the  teeth  developed  in 
the  manner  described. 

Skew  Bevels. 

When   two  axes  or  shafts,  which  have    to  be 
connected  by  bevel  wheels,  do  not  meet  in  direc- 


ON   THE  TEETH   OF   WHEELS.  161 

tion,  it  is  usual,  as  stated  in  the  preliminary  chap- 
ter,*  to  introduce  an   intermediate   bevel  wheel 

Fig.  111. 


with  two  frustra.  But  the  same  object  can  more 
easily  be  accomplished  by  adopting  skew  bevels. 

Let  E  p  q  (fig.  Ill)  be  the  place  of  one  of  the 
two  frustra,  a  its  centre,  and  a  e  the  shortest  dis- 
tance between  the  axis  of  B  p  q,  and  the  axis  of 
the  wheel  to  be  connected  with  it.  Divide  a  e  in 
c,  so  that  a  c  :  e  c  : :  mean  radius  of  A  B  c  :  mean 
radius  of  frustrum  working  with  ABC.  Draw 
c  p  q  perpendicular  to  a  e,  then  c  p  or  c  q  is  the 
line  of  action  of  the  teeth,  according  to  the  di- 
rection in  which  the  teeth  are  laid  out  in  the 
pinion. 

Figure  112  shows  two  wheels  laid  out  in  this 
manner ;  a  e,  as  before,  is  the  eccentricity  or 
shortest  distance  between  the  two  shafts,  and  is 

*  See  page  158,  §  70,  71. 


162 


MACHINERY    OK    TRANSMISSION". 


divided  in  c  proportionally  to  the  mean  radii  of 
the  wheels;  with  centre  a  and  radius  a  c  describe 
a  circle,  and  draw  e  d  perpendicular  to  a  e.  Take 


Fig.  112. 


ON    TllE   TEETH   OF   WHEELS.  163 

df  =  c  e,  then  d  will  be  the  centre  of  the  other 
wheel.  From  centre  d,  with  radius/  d,  describe  a 
circle.  Then  the  directions  of  all  the  teeth  in 
ABC  will  be  tangents  to  the  circle  described 
about  a,  and  the  directions  of  all  the  teeth  in 
D  E  F  will  be  tangents  to  the  circle  described 
about  f.  Fig.  113  shows  two  such  wheels  in 
gear,  the  eccentricity  permitting  the  shafts  to  pass 
each  other. 

Fig.  113. 


The  Worm  and  Wheel. 

By  this  contrivance  the  motion  of  a  screw  is 
communicated  with  great  smoothness  to  oblique 
teeth  on  a  spur  wheel. 

The  section  of  a  screw  through  its  axis  is  pre- 
cisely similar  to  that  of  a  double  rack.  Let  A  B 
be  such  a  section,  and  for  simplicity  suppose  that 
the  form  of  the  threads  of  tho  screw  has  been  de- 


164:  MACHINERY   OF   TRANSMISSION. 

termined  by  one  of  the  rules  already  given  for 
racks.  Then  the  teeth  of  the  wheel  c  D  E  may 
evidently  be  formed  so  as  to  work  with  the  cen- 


Fig.  114. 


tre  section  of  the  screw.  Now  the  effect  of  the 
revolution  of  the  screw  is  precisely  similar  to 
that  of  the  racks,  and  the  sections  of  the  threads 
of  the  screw  will  appear  to  travel  from  end  to 
end,  in  the  same  way  as  a  rack  pushed  forward 
in  the  same  direction.  If,  therefore,  it  is  suffi- 
cient that  the  wheel  teeth  be  in  contact  with  the 
screw  at  one  point  only,  the  teeth  of  the  wheel 
may  be  made  oblique,  but  straight,  the  obliquity 
being  equal  to  the  pitch  of  the  screw.  This  is 
the  usual  practice  of  millwrights.  If,  -however, 
the  teeth  are  required  to  be  in  contact  with  the 


ON    THE   TEETH    OF   WHEELS.  165 

entire  breadth  of  the  tooth,  the  outline  of  the 
tooth  must  vary  in  every  section  of  the  wheel, 
and  the  process  of  describing  these  teeth  becomes 
very  complex.  Practically,  the  .  difficulty  has 
been  overcome  by  first  making  a  pattern  screw  of 
steel,  notched  in  the  threads  to  convert  it  into  a 
cutting  instrument.  The  wheel  is  then  roughly 
cut  out,  and  being  fixed  in  a  frame,  the  screw  is 
used  to  cut  out  the  spaces  between  the  teeth  to 
their  true  form. 

Strength  of  the  Teeth  of  Wheels. 

The  pressure  on  the  teeth  varies  directly  as  the 
horse-power  transmitted  and  inversely  as  the 
velocity  of  revolution.  Thus  if  one  wheel  transmit 
5  horse-power  and  another  10  horse-power  at  the 
same  velocity,  Che  strain  on  the  latter  will  be 
twice  that  on  the  former.  Or,  again,  if  two 
wheels  transmit  the  same  power,  but  one  at  a 
velocity  of  100  feet  per  minute,  and  the  other  at 
only  25  feet  per  minute,  the  strain  on  the  former 
will  be  only  one-fourth  that  on  the  latter. 

Let  v  be  the  velocity  in  feet  per  second,  H  the 
number   of    horse-power    transmitted,    then    the 
total  pressure  on  the  wheels  will  be — 
550  H 

T>     —  -    I 

V 

where  p  is  the  statical  pressure  in  Ibs. 

For  example,  suppose  the  fly-wheel  of  an  en- 


166 


MACHINERY  OF   TRANSMISSION. 


gine  to  be  24  feet  in  diameter,  and  to  work  into  a 
pinion  5  feet  diameter.  And  let  the  work  trans- 
mitted be  150  horse-power.  Then,  if  the  wheel 
makes  25  revolutions  per  minute,  the  periphery 

7o*4:  x  25 
will  move  at  a  velocity  of  -  -  =  31-4    feet 

per  second ;  and  the  statical  pressure  on  the  teeth 


will  be 


550  x  150 
31-4 


=  2627  Ibs. 


In  addition  to  statical  pressure,  however,  a  dif- 
ferent element  has  to  be  taken  into  account,  name- 
ly, the  impacts  due  to  sudden  accelerations  or  re- 
tardations of  speed.  The  allowance  which  must 
be  made  to  prevent  accident  from  this  cause 
varies  exceedingly  in  different  kinds  of  ma- 
chinery. It  is  great  in  the  gearing  of  rolling 
mills  for  instance,  and  in  all  machinery  in  which 
the  strains  are  irregular. 

In  calculating  the  strength  of  the  tooth,  it  has 
been  usual  to  consider  it 
as  a  short  beam  fixed  at 
one  end,  and  having  the 
whole  of  the  pressure  ap- 
plied along  the  extremity 
of  the  tooth.  But  there  is 
a  position  in  which  the 
teeth  may  be  subjected  to 
a  severer  stress  still ;  ow- 
ing to  the  wear  of  brasses 
and  teeth,  we  cannot  calculate  upon  the  strain 


Fig.  115. 

£ 


ON  THE  TEETH  OF  WHEELS.  167 

bearing  always  on  the  whole  breadth  of  the  tooth, 
The  pressure  may  not  only  come  on  to  the  extre- 
mity of  a  tooth,  but  if  any  obstruction  come  in 
between  the  teeth,  it  may  be  thrown  entirely  upon 
one  corner  of  the  tooth.  In  such  a  case  it  may 
be  shown,  by  the  rules  of  maxima  and  minima, 
that  if  E  c  =  C  B,  the  greatest  stress  will  be  near 
the  line  E  B. 

Tredgold  has  expressed  the  strength  of  a  tooth 
on  this  supposition  by  the  formula 

w=s/* 

where  d  is  the  thickness  of  the  tooth.     To  allow 
for  wear,  however,  he  adds  one-third,  so  that 
__f<p(l  —  ff_    fd* 
~6~       "1^25 
In  cast-iron  /=  15,300,  and  hence 


1500 

Or  in  words,  the  thickness  necessary  for  the  tooth 
or  inches  is  equal  to  the  square  root  of  the  stress 
on  the  tooth  in  pounds  divided  by  1500.  Hence 
Tredgold  has  computed  the  following  table,  the 
breadths  of  the  teeth  being  deduced,  on  the  prin- 
ciple that  the  stress  should  not  exceed  400  Ibs. 
per  inch  breadth : — 


168 


MACHINERY   OF  TRANSMISSION. 


TABI.K  OF  THICKNESS,  BREADTH,  AND  PITCH  OF  TEETH  OP 
WHEELS. 


Stress  in  Ihs.  at 
the  pitch  line. 

Thick  ne»s  of  teeth 
in  inches. 

Breadth  of  teeth 
in  inches. 

Pitch  in  inches, 

400 

0-52 

1 

M 

800 

0-73 

2 

1-5 

1,200 

0-90 

3 

1-9 

1,600 

1-03 

4 

2-2 

2,000 

1-15 

5 

2-4 

2,400 

1-26 

6 

2-7 

2,800 

1-36 

7 

2-9 

3,200 

1-46 

8 

3-0 

3,600 

1-56 

9 

3-3 

4,000 

1-64 

10 

3-4 

4,400 

1-70 

11 

3-6 

4,800 

1-78 

12 

3-7 

5,200 

1-86 

13 

3-9 

5,600 

1-93 

14 

4-0 

6,000 

2-00 

15 

4-2 

To  use  this  table  when  the  horses'  power  trans- 
mitted by  the  wheel  are  known,  the  reader  must 
refer  to  the  table  on  page  172. 

Elsewhere  Tredgold  has  given  a  rule  of  the  fol-* 
lowing  description  :  — 


=  f  J  -  for  cast-iron, 

^ 


where  d  is  the  requisite  thickness  of  a  tooth  to 
transmit  a  force  of  H  horses  at  a  velocity  v  feet 
per  second. 

Hence  Tredgold's  last  rule  for  the  thickness  of 
cast-iron  teeth  is  as  follows  —  "  Find  the  number 
of  horses'  power  transmitted  by  the  wheel,  and 


ON  THE  TEETH   OF   WHEELS.  169 

divide  that  number  by  the  velocity  in  feet  per 
second  of  the  pitch  line  of  the  pinion  or  wheel  ; 
extract  the  square  root  of  the  quotient,  and  three 
fourths  of  this  root  will  be  the  least  thickness  of 
cast-iron  teeth  for  the  wheel  or  pinion."  From 
this  he  derives  a  second  rule  for  the  pitch,  which 
manifestly  depends  on  the  thickness  of  the  tooth, 
namely,  multiply  the  thickness  of  the  tooth  by  2'1 
and  the  product  will  be  the  pitch.  The  same  re- 
sult may  be  obtained  from  inspection  of  the  tables 
I  have  given  at  pages  154,  156.  Wooden  teeth 
he  recommends  to  be  made  of  twice  the  thickness 
of  cast-iron  ones.  But  one-and-a-half  times  the 
thickness  is  a  sufficient  allowance. 

A  writer  in  the  "  Engineer  and  Machinists'  As- 
sistant "  deduces  another  but  equally  simple  rule 
for  the  thickness  of  teeth;  he  assumes  the  rela- 
tion 


twhere  t  is  the  thickness  of  the  tooth,  w  the  pres- 
sure on  the  tooth  and  c  a  constant,  depending  on 
the  nature  of  the  material.  Let  then  a  be  the 
strength  of  a  bar  1  inch  long,  1  broad,  and  1  thick. 
Then,  to  support  a  weight  w  by  a  bar  of  a  length 
?,  and  breadth  b, 


a  x 
suppose  the  breadth  of  the  tooth  to  be  fixed  at 

twice  its  length  ; 
15 


170  MACHINERY   OF  TRANSMISSION. 

|~^T          1^" 

JolTST    *K 

Taking  a  =  8000  Ibs.  for  cast-iron,  2  a  =  16,000 
Ibs.,  but  as  this  is  the  breaking  weight,  the  safe 
working-pressure  will  be  only  1600  Ibs.,  and  the 
thickness  of  the  tooth  for  safe  working  will  be  for 
cast-iron :  > 

*==      _^_  =  0-025  Jw. 
^1600  ^ 

Where  w  being  given  in  Ibs.  t  is  found  in  inches. 
Similarly  for  other  materials  he  obtains : 

c  =  -035  for  brass, 
=  -038  for  hard  wood. 

For  example,  in  the  wheel  assumed  at  p.  164, 
w  was  found  to  be  2627  Ibs.  Hence  the  necessary 
thickness  of  the  tooth,  if  of  cast-iron,  would  be 
•025  V2627  =  1-28  inches.  Referring  to  the  tables 
of  the  relation  of  pitch,  etc.,  we  find  that  the  wheel 
must  be  of  2f  inches  pitch,  the  teeth  of  2*1  inches 
length,  and  the  breadth  of  the  wheel  2'1  x  2  =  4'2 
inches  at  the  least.  By  Tredgold's  latter  rule,  the 
thickness  of  the  teeth  for  the  same  wheel  would 

be  t  =  f  J          =  1*41  inches :  the  pitch 

=  2'1  x  1-41  =  3-0  inches,  and  the  breadth 

2627 
==  400          *  m 


ON  THE   TEETH   OF   WHEELS.  171 

550  H 

Bearing  m  mmd  that  w  =  —  —  ,  where  H  is  the 

v 

maximum  horse-power  transmitted,  and  v  the  velo- 
city of  the  pitch  line  of  the  wheel  in  feet  per  se- 
cond, we  may  give  these  formulae  in  a  more 
convenient  form  : 


Where  x  =  0*587  for  cast-iron, 
"       =  0-821  for  brass, 
«      =0-891  for  wood. 

Conversely,  if  a  wheel  having  teeth  t  inches 
thick  be  given,  the  horse  power  it  is  capable  of 
transmitting  is  given  by  the  formula  : 

fv 

H==^- 

Where  x*  =  0-344  for  cast-iron, 
«        =  0-674  for  brass, 
«        =  0-795  for  wood. 

From  the  following  table  the  pressure  at  other 
velocities,  and  with  another  amount  of  horse-power, 
may  be  obtained  by  interpolation,  remembering 
that  the  pressure  varies  inversely  as  the  former, 
and  directly  as  the  latter.  To  this  we  have  ap- 
pended another  table,  giving  the  horses'  power, 
which  can  be  safely  transmitted  by  wheels  of  dif- 
ferent pitches  when  proportioned  according  to  the 
table  at  page  154.  The  last  of  these  tables  has 
been  calculated  on  the  assumption  that  400  Ibs. 
per  inch  is  the  greatest  working  stress  which  is 
consistent  with  durability  in  ordinary  cases. 


172  MACHINERY   OF   TRANSMISSION. 


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ON   THE  TEETH   OF   WHEELS. 


173 


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174 


MACHINERY   OF   TRANSMISSION. 


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STRENGTH  AND   PROPORTIONS   OF  SHAFTS.    175 

CHAPTEK  III. 

ON   THE   STRENGTH   AND   PROPORTIONS   OF   SHAFTS. 

THE  system  of  transmitting  power  from  a  com- 
mon centre  to  a  large  number  of  machines,  at 
some  distance,  is  comparatively  modern.  In  the 
operations  of  spinning  and  weaving  by  a  consecu- 
tive series  of  machines,  placed  in  rows,  shafting 
became  essential  for  distributing  the  power  of  the 
common  prime  mover.  At  first,  the  machines 
were  brought  as  close  to  the  prime  mover  as  pos- 
sible ;  and  the  early  construction  of  mills — when 
the  water-power  was  divided  into  separate  falls — 
must  be  fresh  in  the  recollection  of  many  persons 
now  living.  In  some  cases,  before  the  introduc- 
tion of  the  steam  engine,  it  was  the  custom  to 
have  a  separate  water-wheel  to  every  machine, 
thus  splitting  up  the  power  into  as  many  parts  as 
there  were  machines,  or  pairs  of  machines,  to 
drive.  In  process  of  time,  it  was  found  more  con- 
venient, on  the  score  of  economy,  to  husband  the 
water  and  concentrate  the  prime  movers;  hence 
one  Urge  water-wheel  was  constructed,  around 
which  the  machinery  was  arranged,  either  in  rows 
or  otherwise  as  best  suited  the  work  to  be  per 
formed. 

This  principle,  of  the  concentration  of  the  mo- 
tive power,  destroyed  the  old  system  of  separate 


176  MACHINERY   OF  TRANSMISSION. 

buildings,  and  led  to  the  employment  of  a  large 
number  of  machines  for  the  various  processes  of 
manufacture  in  one  building.  From  this  we  de- 
rive the  Factory  system,  in  which  any  number  of 
processes  are  carried  on,  the  machinery  being  dis- 
tributed over  the  different  floors  of  a  large  build- 
ing, and  receiving  motion  from  a  single  prime 
mover  at  a  convenient  distance.  In  this  way,  the 
power  is  conveyed  by  lines  of  shafting  coupled 
together  in  lengths,  adapted  to  the  bays  or  divis- 
ions of  the  building.  At  first,  the  buildings  were 
short,  and  shafting  of  great  length  was  not  re- 
quired ;  gradually,  more  and  more  machines  were 
concentrated  in  the  same  building,  and  shafting  of 
200  or  300  feet  in  length  became  necessary.  To 
show  to  what  an  extent  this  system  has  been  car- 
ried, it  may  be  mentioned  that,  in  the  large  mills  at 
Saltaire,  the  shafting,  if  placed  in  a  single  line, 
would  extend  for  a  distance  of  more  than  two 
miles.  This  progress  has  been  chiefly  due  to  the 
introduction  of  the  steam  engine,  in  place  of  water- 
wheels,  because  the  available  power  is  no  longer 
limited  by  the  circumstances  of  the  locality  in 
which  the  mill  is  placed. 

This  concentration  of  a  great  number  of  ma- 
chines in  one  building  is  peculiar  to  the  Factory 
system  ;  and  in  the  present  highly-improved  state 
of  mechanical  science  and  its  application  to  the 
production  of  textile  fabrics,  it  has  become  essen- 
tial to  economy  in  the  manufacturing  processes, 


STRENGTH   AND   PROPORTIONS   OF   SHAFTS.     177 

that  they  should  be  carried  on  in  the  same  build- 
ing. Spinners  and  manufacturers  are  fully  aware 
of  the  advantages  peculiar  to  this  system  of  con- 
centration, so  much  so,  that  out  of  what  would 
formerly  have  been  considered  a  mere  fractional 
saving,  large  profits  and  large  fortunes  are  now 
made.  In  fact,  the  amalgamation  of  the  different 
processes  under  one  management  and  under  one 
roof,  gave  rise  to  the  shed  system,  where  the 
operations  of  the  manufacture  of  cotton  are  carried 
on  under  what  is  called  the  "sawtooth"  roof,  in 
order  to  bring  the  whole  on  the  ground-floor 
under  one  inspection. 

1.  The  Material  of  which  Shafting  is  constructed. 

The  selection  of  the  material  for  shafting  is  of 
great  importance,  and  the  uses  to  which  it  is  to  be 
applied  require  careful  consideration.  Formerly 
wood,  with  iron  hoops  and  gudgeons,  was  univer- 
sally employed ;  then  cast-iron  was  introduced 
and  subsequently  wrought-iron  has  in  most  cases 
superseded  both.  Wood,  indeed,  has  become 
obsolete ;  but  cast-iron  is  as  good  as,  if  not  superior 
to,  wrought-iron,  in  certain  cases.  The  main  and 
vertical  shafts  of  a  mill  are  generally  of  cast-iron, 
both  on  account  of  its  cheapness,  and  its  high  re- 
sistance to  torsion.  The  vertical  shafts,  which 
convey  the  power  from  the  first  motion  wheels  to 
the  different  rooms  of  the  mill,  are  more  rigid 
and  less  subject  to  vibration  when  of  cast- iron ; 


178  MACHINERY   OF   TRANSMISSION. 

even  the  main  horizontal  shafting,  when  of  large 
dimensions,  is,  if  substantially  fixed,  quite  as  good, 
when  of  the  same  material,  and  much  cheaper 
than  wrought-iron.  Where  the  shaft  is  exposed 
to  impact,  or  any  irregularity  of  force,  wrought- 
iron  has  the  superiority ;  but  in  other  cases,  when 
the  castings  are  sound  and  good,  cast-iron  may  be 
employed  with  perfect  safety. 

The  dimensions  required  for  a  shaft,  transmit- 
ting any  given  force,  will  depend  on  the  resistance 
of  the  material  of  which  it  is  composed.  Conse- 
quently, the  selection  of  material  must  be  deter- 
mined by  the  necessity  for  strength.  Shafts  may 
be  considered  as  subject  to  two  forces :  a  force 
producing  simple  flexure,  arising  from  their  own 
weight,  the  weight  of  the  wheels  and  pulleys-,  and 
the  strain  of  the  belts ;  and  a  twisting  force  or 
torsion,  arising  from  the  power  transmitted.  If 
the  flexure  be  great,  the  brasses  will  be  much 
worn,  vibration  becomes  considerable,  and  the 
disintegration  of  the  machinery  goes  on  in  an 
accelerating  ratio ;  it  is  therefore  necessary  to  pro- 
portion shafting  to  the  simple  weight  and  direct ' 
transverse  strain  it  has  to  sustain,  so  as  to  reduce 
the  flexure  within  exceedingly  narrow  limits. 
In  addition  to  this,  the  shafting,  having  to  trans- 
mit a  torsive  force,  must  at  least  be  capable  of 
transmitting  it  without  danger  of  rupture.  In 
long  and  light  shafting  the  tendency  to  flexure  is 
usually  greater  than  that  to  rupture  by  torsion ; 


STRENGTH  AND   PROPORTIONS   OF   SHAFTS.     179 

the  former  consideration  will  therefore  determine 
the  size  of  the  shaft.  In  short  axles,  etc.,  the  dan- 
ger from  flexure  almost  disappears,  and  the 
strength  of  the  shaft  is  determined  by  its  resist- 
ance to  torsion  only.  In  all  cases  both  conditions 
must  be  complied  with,  if  security  and  permanence 
are  to  be  obtained. 

2.  Transverse  Strain. 

Resistance  to  rupture.  The  general  formula  for 
resistance  to  rupture,  in  the  case  of  a  bar  or  beam 
supported  at  each  end  and  loaded  in  the  centre,  is 

adc 
w=-r...(l), 

where  w  is  the  load  in  the  centre,  a  the  area  of  a 
section  of  the  bar,  perpendicular  to  the  length  ;  d 
the  depth  of  the  bar,  and  I  its  length.  In  this 
case  c  is  derived  from  experiment,  and  is  constant 
for  similar  bars  or  beams. 

For  rectangular  bars  this  formula  becomes, 

cbd* 
w=—  ...(2). 

where  b  is  the  breadth  and  d  the  depth. 

The  value  of  c,  for  rectangular  bars  found  by 
Mr.  Barlow,  for  various  materials,  is  given  in  the 
following  table.  In  applying  these  numbers  to 
calculations,  it  must  be  remembered  that  a  and  d 
are  to  be  taken  in  inches,  and  I  in  feet ;  then  c 
the  centre  breaking-weight,  is  found  in  Ibs. 

v 


180         MACHINERY  OF  TRANSMISSION. 

When  the  beam  is  supported  at  one  end  and 
loaded  at  the  other,  the  formula  is 
cbd* 


Value  of  c  for  different  Materials. 

Ibs. 
English  Malleable  Iron  ...............  2050 

Cast  Iron  ...........................  2548 

Oak  .................................     400 

Canadian  Oak  ......................     588 

Ash  ................................     675 

Pitch  Pine  .........................     544 

Bed  Pine  ............................    447 

KigaFir  ............................    376 

Mar  Forest  Fir  .........  «  ............     415 

Larch  .............................     280 

In  my  own  experiments*  I  found  the  value  of 
c  for  cast-iron  to  range  from  1606  to  2615,  the 
mean  value  being  about  2050,  as  given  above  for 
malleable  iron.  Wrought-iron  ranges  from  the 
value  given  above  to  3000  Ibs. 

For  cylindrical  shafts  supported  horizontally 
the  ultimate  resistance  to  rupture  is  about 

1500Q*, 

W  =  -  j  -  for  wrought-iron, 

19000  d*  „  . 
=  -  j  -  for  cast  -iron, 

I 

where  W  is  the  centre-breaking  weight  in  Ibs.,  d 

*  On  the  Application  of  Cast  and  Wrought-iron  to  Building 
Purposes,  p.  74,  et  seq. 


STRENGTH  AND  PROPORTIONS  OF  SHAFTS.    181 

the  diameter,  and  I  the  length  between  supports 
in  inches,  the  shaft  being  supported  at  the  ends 
and  loaded  in  the  middle. 

If  the  cylindrical  shaft  be  loaded  at  one  end 
and  supported  at  the  other,  these  formulae  become 

7500  d>  £ 
W  =  -  7  -  for  wrought-  iron, 

9500  d*  , 
==  -  =  -  for  cast-iron. 

• 

If  a  beam  be  uniformly  loaded  over  its  entire 
length  it  will  sustain  twice  the  load  that  would 
break  it  if  placed  at  the  centre. 

If  the  load  be  placed  at  any  point  intermediate 
between  the  centre  and  the  ends,  the  breaking 
weight  may  be  found  by  the  following  rule  :  — 
Divide  four  times  the  product  of  the  distance  in 
feet,  of  the  weight  from  each  bearing,  by  the 
•whole  distance  in  feet,  and  the  quotient  may  be 
substituted  for  I  in  the  formulae  above.  That  is, 
if  x  and  y  be  its  distances  in  feet  from  the  two 
bearings  respectively  ; 


"(*+?)• 

From  these  rules  the  strength  of  shafts  may  be 
calculated,  in  all  the  cases  of  ordinary  practice, 
where  the  tendency  to  transverse  fracture  has  to 
be  guarded  against,  making  the  actual  strength  at 
least  five  to  ten  times  the  strain  to  be  carried. 
In  shafting,  however,  it  is  not  usually  the  trans- 
16 


182  MACHINERY   OF   TRANSMISSION. 

verse  rupture,  but  the  flexure  produced  by  lateral 
stress,  which  limits  the  size  of  the  shaft  ;  —  stiffness 
in  fact  becomes,  in  these  cases,  a  more  important 
element  than  strength. 

The  following  formula  has  been  given  for  the 
deflection  of  bars  or  beams  loaded  at  the  centre 
and  supported  at  the  ends  :  — 
Let,     d  be  the  depth  in  inches  ; 
b  the  breadth  in  inches  ; 
L  the  length  between  supports  in  feet  ; 
w  the  load  in  Ibs.  ; 

8  the  deflection  at  the  centre  in  inches  ; 
M  the  modulus  of  elasticity  ; 

ubd'8 
then  :  —  w  =  •  .on  T  a   ,'  and  if  o  =  a,  — 


.........  (4). 


Or,  in  words,  multiply  the  product  of  the  load 
in  Ibs.,  and  the  cube  of  the  length  in  feet,  by  432, 
and  divide  by  the  product  of  the  modulus  of  elas- 
ticity and  the  deflection  assumed  in  inches  ;  the 
fourth  root  of  the  quotient  will  be  the  side  of  a 
shaft  or  square  section  which  would  deflect  s 
inches  with  a  weight  of  w  Ibs.  placed  at  its 
centre.* 

*  Engineer  and  Machinist's  Assistant,  p.  135,  from  which 
formulae  (4),  (5),  (6),  to  (11),  and  (23),  in  their  present 
convenient  form  for  practical  use,  have  been  quoted.  The 
fundamental  formula,  however,  is  due  to  Young  (Nat. 


STRENGTH   AND   PROPORTIONS   OF  SHAFTS.    183 

The  following  table  gives  the  values  of  the 
modulus  of  elasticity  for  various  materials  :  — 

Modulus  of  elasticity  la  Ibs. 

Cast-iron  ........  13,000,000  to  22,907,000 

"       "      mean  ...............  17,000,000 

Malleable  iron.  ..24,000,000  to  29,000,000 
Steel  ............  29,000,000  to  42,000,000 

Brass  .........................  8,930,000 

Tin  ...........................  4,608,000 

Ash  ..........................  1,600,000 

Beech  ..........................  1,353,600 

Eed  pine,  mean  ................  1,700,000 

Spruce,  mean  ..................  1,600,000 

Larch  ...............  900,000  to  1,360,000 

English  oak.  .'.  .....  1,200,000  to  1,750,000 

American  oak  .................  2,150,000 

For  a  cylindrical  shaft,  the  same  formula  will 
apply  with  another  constant.  I  am  not  aware 
that  this  has  been  experimentally  ascertained,  but 
it  has  been  given  approximately  as  734.  Hence, 
for  cylindrical  shafts, 


M5  N       M8 

In  the  work  just  quoted,  these  formulae  have 
been  simplified,  by  fixing  a  maximum  value  for  8, 
the  deflection.  The  writer  assumes  that,  with 
shafting,  the  deflection  ought  never  to  exceed  f$w 
of  an  inch  for  every  foot  length  of  the  shaft. 
Substituting  this  value,  and  also  the  numerical 

Philos.,  vol.  ii.,  art.  326),  and  to  Tredgold  (Strength  of  Cast- 
iron,  p.  208). 


184:  MACHINERY   OF   TRANSMISSION. 

value  of  the  modulus  of  elasticity,  he  obtains  the 
following  formulae  :  — 

1.  For  wood,  —  taking  M  generally  =  1,500,000, 

and  8  =  inches. 


Then,  for  square  shafts,  d  being  the  depth  of 
the  side  of  the  square  — 

L2w 
*~  -M-  -  ^ 

And  for  round  shafts,  d  being  the  diameter  in 
inches  — 

L"W 

*—n-~<fr 

2.   For  cast-iron  —  taking  M  =  18,000,000  Ibs. 
and  L  as  before  — 

L2W 

For  square  section,  d4  =    .10    ...  (8). 

" 


For  round  section,  d4  =  -JTTTT-  •••  (9). 

~ 


3.   For  wrought-iron  —  taking   M  =  24,500,000 
Ibs.  and  3  as  before  — 

L2w 
For  square  section,  d4  =  -Vo-  •••  (10). 


L2  W 

For  round  section,  d4  =          -  ...  (11). 

OOtt 

By  transposition,    the   formula3    given   above 
become,  — 


STRENGTH   AND   PROPORTIONS   OF   SHAFTS.    185 

For  wood  — 

Square  section,  L  =  J-—  ...  (12). 
w 

35  d4 


Kound  section,  L  =     j          »•  (13). 
w 

20  # 
w=    —-  ...(14). 


For  cast-iron  — 


Square  section,  L  =  .-•  (15). 


Eound  section,  L=J240fl?<.  ..(17). 

^ 


w 

240^ 


For  wrought  iron  — 

Square  section,  L  =J567^  ...(19). 


w= 


Eound  section,  L  ==  ...(21). 

~     w 


334  d4 
w=  - 


16* 


186  MACHINERY   OF   TRANSMISSION. 

When  the  weight  is  uniformly  distributed  over 
the  length  of  the  shaft,  the  general  formula  is 

270  L'W*  27017^ 


Substituting  in  this  equation  the  same  values 
of  M  and  8  as  before,  we  obtain  the  following  for- 
mulae :  — 

L2  w 

For  wood  —          d4  =  -^-  for  square  shafts. 

L2  w 
d*  =  —j.f.  for  round  shafts. 

O2J 

L2  W 

For  cast-iron  —    d4  =  ^i*  for  square  shafts. 
666 

L2  w 

d4  =  5^5  for  round  shafts. 
ooo 

L2  W 
For  wrougld-iron  d4  =  q^y-  for  square  shafts. 


L2  w 
d*  =      r  for  round  shafts. 


The  following  tables  for  cast  and  wrought-iron 
round  shafting,  are  calculated  from  the  formulae 
(9)  and  (11)  for  weights  placed  at  the  centre  of  a 
shaft  supported  at  each  end.  In  using  them  for 
cases  in  which  the  weight  is  distributed  along^its 
length,  as  in  the  case  of  -the  weight  of  the  shaft 
itself,  it  must  be  remembered  that  a  distributed 
weight  produces  f  ths  of  the  deflection  of  the  same 
weight  placed  at  the  centre. 


STRENGTH   AND   PROPORTIONS   OF   SHAFTS.    187 


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188 


MACHINERY   OF   TRANSMISSION. 


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STRENGTH  AND  PROPORTIONS  OF  SHAFTS.  189 


From  the  foregoing  it  will  be  seen,  that  the 
weights  given  in  the  tables  are  correct  indications 
of  the  load  required  in  the  centre  to  produce  a 
deflection  of  the  jVau  °f  ^ne  length  of  the  shaft.* 
This  fraction  is  not  however  the  universal  stand- 
ard among  millwrights;  on  the  contrary,  there 
appears  to  be  no  recognized  standard  in  practice, 
by  which  the  deflection  from  a  given  weight  can 
be  ascertained,  and  although  T515?j  may,  in  many 
cases,  give  a  larger  area  with  increased  weight,  in 
shafts  that  are  not  heavily  loaded  in  the  middle, 
nevertheless  it  is  important  that  the  shafts,  when 
loaded  as  above,  should  not  bend  more  than  j2Va 
of  their  length.  In  cases  where  the  load  is  light 
and  equally  distributed,  lighter  and  smaller  shafts 
would  suffice. 

The   following  tables   give   the  deflection    of 
cylindrical  shafts  with  their  own  weight : — 

TABLE  3. — DEFLECTION  ARISING  FROM  THE  WEIGHT  OF  THE 
SHAFT.     CAST-IRON  CYLINDRICAL  SHAFTS. 


Diameter  of  Shaft  in  Inches. 

Length  between 

feet. 

1 

2 

4 

6 

8 

10 

12 

14 

16 

ins. 

ins. 

ing. 

ins. 

ins. 

ins. 

ins. 

ins. 

ins. 

ins. 

5 

•004 

•001 

•000 

•000 

•000 

•000 

•000 

•000 

•000 

10 

•067 

•017 

•004 

•002 

•001 

•001 

•001 

•000 

•000 

15 

•338 

•085 

•021 

•009 

•005 

•003 

•002 

•002 

•001 

20 

1-067 

•267 

•067 

•029 

•017 

•Oil 

•007 

•005 

•004 

25 

2-603 

•651 

•163 

•073 

•041 

•026 

•018 

•013 

•010 

*  This  standard  is  the  one  assumed  by  Tredgold  (Strength 
of  Cast-Iron,  p.  210). 


190 


MACHINERY   OF   TRANSMISSION. 


TABLE  4. — DEFLECTION  ARISING  FROM  THE  WEIGHT  OF  THE 
SHAFT.     WROUGHT-IRON  CYLINDRICAL  SHAFTS. 


Length  between 
bearings  in 
feet. 

Diameter  of  Shaft  in  Inches. 

1 

2 

4 

ins. 

6 

8 

10 

12 

14 

16 

in« 

Ins. 

ins. 

ins. 

ins. 

ins. 

ins. 

ins. 

ins. 

5 

•003 

•001 

•ooo 

•ooo 

•ooo 

•ooo 

•0)0 

•ooo 

•ooo 

10 

•050 

•013 

•003 

•001 

•001 

•001 

•ooo 

•ooo 

•ooo 

15 

•256 

•064 

•016 

•007 

•004 

•003 

•002 

•001 

•001 

20 

•808 

•202 

•051 

•022 

•013 

•008 

•005 

•004 

•003 

25 

1-972 

•493 

•123 

•055 

•031 

•020 

•013 

•010 

•008 

The  above  tables  clearly  indicate  the  deflection 
of  shafts  of  different  lengths  by  their  own  weight, 
and  will  be  a  guide  to  the  millwright  in  calculat- 
ing the  distance  of  the  bearings  between  which 
they  revolve.  It  is  important  in  shafting,  when 
extended  in  long  ranges,  that  there  should  not  be 
any  serious  deflection,  either  from  the  weight  of 
the  shaft,  or  lateral  stress ;  I  have  always  found 
that  a  stiff  shaft,  although  heavier  in  itself,  is 
lighter  to  retain  in  motion  than  a  smaller  one 
which  bends  to  the  strain. 

3.   Torsion. 

In  addition  to  the  lateral  flexure  from  trans- 
verse forces,  shafting  is  subjected  to  a  wrenching 
or  twisting,  from  the  power  transmitted  acting 
tangentially  to  its  circumference.  This  causes 
one  end  of  the  shaft  to  revolve  in  relation  to  the 


STRENGTH   AND   PROPORTIONS   OF   SHAFTS.    191 

other  end,  through  a  smaller  <5r  greater  angle, 
known  as  the  angle  of  torsion,  and  if  sufficient 
force  be  applied,  this  angle  increases  till  the 
resistance  of  the  material  is  overcome,  and  the 
shaft  gives  way. 

Coulomb  laid  the  basis  of  our  knowledge  of  the 
resistance  to  torsion  of  cylindrical  bodies,  and  he 
verified  his  theoretical  deductions  by  admirably- 
contrived  experiments,  on  a  small  scale.  He 
showed  that  in  wires  where  "the  diameter  is  small 
in  relation  to  the  length,  the  angles  of  torsion 
are  in  proportion  to  the  length,  and  reciprocally 
proportional  to  the  moment  of  inertia  of  the  base 
of  the  cylinder  in  relation  to  its  centre.  He  also 
discovered  that  each  wire  acquired  a  permanently 
acceleration-varying  torsion,  according  to  the  de- 
gree in  which  it  departed  from  its  primitive  posi- 
tion, and  that  these  permanent  torsions  have  no 
fixed  relation  to  the  temporary  torsions,  coexist- 
ing with  the  application  of  the  moving  force. 
With  the  same  wire  he  found  the  torsion  to  be  in 
proportion  to  the  force  applied;  with  the  same 
length  and  force  inversely  as  the  fourth  power  of 
the  diameter. 

These  deductions  are  expressed  by  the  follow- 
ing formula  :— 

2R       wZ 

6=  ~  ~    X    — r 

*G        r4 

where  o  is  the  angle  of  torsion,  r  the  radius,  and  2 
the  length  of  the  wire,  R  the  leverage  at  which 


192  MACHINERY  OF  TRANSMISSION. 

the  weight  W  acts,  and  G  the  modulus  of  torsion 
for  the  material  ;  being  about  f  ths  of  the  modulus 
of  elasticity. 

In  1829  a  paper  was  communicated  to  the 
Eoyal  Society  by  Mr.  Bevan,  containing  experi- 
mental determinations  of  the  modulus  of  torsion 
for  a  large  number  of  substances,  of  which  the 
most  important  are  given  below. 

Let  8  be  the  deflection  of  a  prismatic  shaft  of  a 
given  length  I  when  strained  by  a  given  force  w 
in  Ibs.,  acting  at  right  angles  to  the  axes  of  the 
prism  and  at  a  leverage  r  ;  let  d  be  the  side  of  the 
square  section  of  the  shaft,  I,  r,  8,  d  being  in  iiichs. 


where  T  is  the  modulus  of  elasticity  in  the  follow- 
ing table. 

If  the  transverse  section  of  the  prism  be  a 
parallelogram,  let  b  be  the  breadth  and  d  the 
depth,  then  Mr.  Bevan  gives  the  formula  — 


_ 


2bd*T 

If  the  torsion  be  required  in  degrees  (A),  then  let 
.  =  57-29578, 

rplw 
A  =  —p  —  ,  for  square  shafts. 

For  example, 

rlw 

wrought-iron  and  steel, 


STRENGTH   AND  PROPORTIONS   OF   SHAFTS.    193 

rlw 
=  16600*  f°r  cast-iron' 

A  very  careful  experimental  study  of  the  effect 
of  torsion  on  various  materials  has  been  made  by 
Mr.  M.  G.  Wertheim,  and  was  presented  to  the 
Academic  des  Sciences  in  1855.  The  general  re- 
sults at  which  he  has  arrived  may  be  stated  as 
follows : — 

1.  The  total  angle  of  torsion  consists  of  two 
parts,  of  which  one  is  purely  temporary,  whilst 
the  other  persists  after  the  force  has  ceased  to  act. 
It  is  not  possible  to  assign  the  limit  at  which  the 
permanent  torsion  begins  to  be  sensible,  nor  has 
it  any  fixed  relation  to  the  temporary  torsion  ;  it 
augments  at  first  very  slowly,  afterward  more 
rapidly,  till  the  bar  breaks.* 

*  We  have  many  practical  instances  of  this  tendency  to 
rupture  which  at  first  appear  only  temporary,  but  a  con- 
tinuation of  the  same  action,  particularly  in  long  ranges 
of  shafts,  in  process  of  time,  developes  itself  in  the  form  of 
a  permanent  deterioration  which  ultimately  leads  to  frac- 
ture. This  was  strikingly  exemplified  in  a  range  of  shafts, 
220  feet  long,  tapering  from  three  inches  diameter  at  the 
driving  end,  to  two  inches  diameter  at  the  other. 

The  work  done  by  these  shafts  was  uniform  throughout, 
but  it  was  soon  found  that  the  shaft  had  made  nearly  1-16 
revolutions  at  the  driven  end  of  the  room,  before  it  began 
to  move  at  the  other.  The  result  was  a  continued  series 
of  jerks  or  accelerated  and  retarded  motion,  injurious  to 
the  machinery,  and  destructive  to  the  work  it  had  to  per- 
form. It  was,  moreover,  injurious  to  the  shafts,  particu- 
larly in  the  middle,  where  the  twist  was  severely  felt,  and 
17 


194 


MACHINERY   OF  TRANSMISSION. 


TABLE  5. — VALUES  or  MODULUS  OF  TORSION  ACCORDING  TO 
MR.  BEVAN. 


Material. 

V  08 

oc  ac 

Modulus  of 
torsion. 

Ash           .        ..      .-- 

20',300 

Beech 

— 

21,243 

Elm 

— 

13,500 

Scotch  fir  . 

— 

13,700 

Hornbeam 

•86 

26,400 

Larch 

•58 

18,967 

English  oak 

— 

20,000 

Memel  pine 

— 

15,000 

American  pine  . 

— 

14,750 

Teak 

— 

16,800 

Old  and  parti- 

— 

ally  decayed. 

Teak,  African    . 

— 

27,300 

Iron,  English  wronght 
Steel 

— 

1,775,000 
1,753,000 

(Mean.) 
(Mean.) 

Iron  (cylindrical) 

— 

1,910,000 

, 

— 

1,700,000 

(square)     . 

— 

1,617,000 

"             .        . 

— 

1,667,000 

w 

— 

1,951,000 

Cast-iron  . 

— 

940,000 

"... 

— 

963,000 

"... 

— 

952,000 

"... 

— 

951,600 

(Mean.) 

Bell  metal 

— 

818,000 

2.  The  temporary  angles  are  not  rigorously  pro- 
portional to  the  moments  of  the  forces  applied. 

3.  The  mean  angles  of  torsion  are  not  rigorously 
proportional  to  the  length  of  the  bar,  increasing, 

would  have  led  to  rupture,  but  from  the  circumstance 
that  they  had  to  be  renewed  with  a  stiffer  and  stronger 
range. 


STRENGTH   AND   PROPORTIONS   OF   SHAFTS.    195 

although  very  slightly,  in  proportion  to  the  length, 
as  the  bars  are  made  shorter. 

4.  The  interior  cavity  of  all  hollow  homogene- 
ous bodies  diminish  by  torsion,  and  this  diminu- 
tion is  proportional  to  the  length  and  to  the  square 
of  the  angle  of  torsion  for  unity  of  length. 

5.  For  cylindrical  bodies  Mr.  Wertheim  gives 
the  following  formulae  : — 

Let  *•  be  the  mean  temporary  angle  of  torsion, 
for 

p  =  1  kilogramme,  and 
1=  1  metre; 
p  =  the  sum*  of  the  two  weights  producing 

torsion   and  constituting  a  couple  in 

kilogrammes ; 
B  =  the  leverage  at  which  the  weight  p 

acts: 
I  =  length  of  the  bar  subject  to  torsion,  in 

millimetres ; 
r  =  the  exterior  radius  of  the  section  of 

the  bar,  in  Millimetres  ; 
ri=the  interior  radius  of  hollow  bars,  in 

millimetres ; 

E  =  the  modulus  of  elasticity  of  the  mate- 
rial  obtained   from   experiments    on 

tension. 

*  In  Mr.  Wertheim's  experiments  equal  weights,  acting 
in  opposite  directions  at  the  same  leverage,  were  hung  one 
on  each  side  of  the  bar,  subjected  to  torsion., 


196 


MACHINERY  OF   TRANSMISSION. 


Then  for  solid  bars : — * 

_  16^       180      p R       l_ 
'  IT   '    ~S    '  ~-R    '    r* 
and  for  hollow  cylinders 

16_      180     £B      _[_ 
~3    '    "«*    *.,•*.'  r^r4 
In  the   following   experiments,  p  =    1    kilo 
gramme,  R  =  247'5  millimetres,  2  =  1000  milli 
metres. 

RESUME  OF  EXPERIMENTS  ON  CYLINDERS  OF  CIRCULAR 
SECTION. 


Material 

Radius 

Coefficient 
of  elasti- 

Mean angle  of  torsion. 

r. 

city,  E. 

By  formula. 

Byexperiment. 

mm. 

o       /       // 

o      i          >r 

1 

Iron 

8-220 

17,805 

0  17  46-1 

0  17  52-1 

2 

Iron 

5-501 

« 

1  28    0-8 

1  26  31-3 

3 

Cast  steel 

5-055 

19,542 

1  53  12-0 

1  51  13-4 

4 

Copper     . 

5-031 

9,395 

3  59  59-1 

3  54     6-0 

0 

Glass 

3-535 

6,200 

24  51  56-0 

24  15  34-7 

6 

Glass 

3-4225 

<« 

28  18    2-0 

28  30  14:0 

The  accordance,  in  these  tables,  between  the 
formulae  and  the  experiments  is  very  satisfactory, 
especially  considering  that  the  value  of  E  cannot 
be  determined  with  perfect  accuracy.  The  errors 
do  not  generally  exceed  ^Oth,  and  the  observed 

*  The  above  formulae  may  be  lased  with  English  mear- 
ures,  E  being  taken  from  English  tables,  if  p  be  given  iu 
Ibs.  and  r,  I,  and  R  in  inches. 


STRENGTH   AND   PROPORTIONS   OF  SHAFTS.    197 


angles  are  smaller  than  those  found  by  calculation 
except  in  the  case  of  the  cylinders  9,  53,  and  54. 

RESUME  OF  EXPERIMENTS  ON  THE  TORSION  OF  HOLLOW 
CYLINDERS  OF  COPPER. 


ft 

R* 

Coefficient  of 

Angle  of  torsion  (¥). 

l^ 

•Si- 

elasticity 

fri    5 

£  § 

from  tension 

r 

H 

H 

(B). 

By  formula. 

By  experiment. 

53 

11,525 

10,021 

10.917 

o      /        n 

0  17  30-2 

o      /        n 

0  20    0-6 

54 

7,082 

4,955 

10.444 

1  12  18-3 

1  16  52-9 

55 

5,047 

30,315 

10,276 

4    9    4-0 

4    6  54-7 

7 

5,602 

24,665 

9,665 

2  37  40-4 

2  33  38-2 

8 

45,605 

2,478 

9,855 

6  11  10-3 

6    0  53-8 

9 

36,955 

2,471 

10,645 

15    9  14-4 

15  42  37-3 

For  bars  of  elliptical  section  M.  Wertheim  has 
deduced  the  formula 

8       180     j9R      Z(<3  +  <£) 
:~IF  '  "77  '  IF  '     c\c\ 

where  el  and  Cj  are  the  two  semiaxes  of  the  ellipse, 
the  other  letters  remaining  as  before. 

RESUME  OF  EXPERIMENTS  ON  THE  TORSION  OF  ELLIPTICAL 
BARS. 


Semiazes. 

Coefficient  of 

Mean  angle  of  torsion  W). 

Material. 

elasticity  by 
tension 

Cl 

c, 

(E.) 

By  formula. 

By  exper't. 

mm. 

mm. 

o   i    n 

o   i      n 

11 

Cast  steel... 

7,106 

3,697 

19,085 

2  13  56-7 

2  10  55-4 

12 

" 

9,900 

25,075 

« 

4  18    0-1 

4  13  18-2 

13 
14 

Copper  

7,062 
9,875 

3,669 
2,498 

9,634 

4 

4  32  56-7 
8  38  11-2 

4  30  41-2 
8  54  33-9 

17* 


198  MACHINERY   OF   TRANSMISSION. 

For  bars  of  rectangular  section  the  formula  be- 
comes 

_180    1    ^R    j(o'  +  y) 
=  "«•"  '  2  '  "K"  *       ~^¥~ 

But  it  is  necessary  to  apply  a  coefficient  of  cor- 
rection c  to  the  calculated  angle  such  that  if  -*-1  be 
the-  calculated  angle  of  torsion,  and  **  the  angle 

found  by  experiment,  then  c  =  — ,  .     This  coeffi- 
cient varies  with  the  ratio  r  of  the  sides  of  the  bar. 

Thus  when  I  =  500  millimetres,  and  the  section 
was  36  milimetres  square. 

£ 1  2  4         8 

Value  of  coefficient    0-8971  0-9617  0-9520  0-9878 

It  varies  also  with  the  ratio  r  and  with  the  mo- 

0 

ment  of  the  couple  p  R. 

For  the  ultimate  resistance  of  cylindrical  shafts 
to  rupture  by  torsion,  Professor  W.  J.  M.  Rankine 
gives  the  following  formula  :* 

Let  I  denote  the  length  in  inches  of  the  lever, 
such  as  a  crank,  at  the  end  of  which  a  wrenching 
or  twisting  force. is  applied  to  an  axle.  Let  w  be 
the  working  load  in  pounds,  multiplied  by  a  suit- 

*  Manual  of  Applied  Mechanics,  p,  355.  Manual  of 
Steam  Engine,  p.  78. 


STRENGTH   AND   PROPORTIONS  OF  SHAFTS.    199 

able  factor  of  safety  (usually  six)  ;  then 

wZ=M 

is  the  wrenching  moment  in  inch  pounds. 
For  a  solid  axle  let  h  be  its  diameter  ;  then 

" 


»-./*,»**.=  "I— 

51  "J 


For  a  hollow  axle  let  h^  be  the  external,  and«A, 
the  internal  diameter  in  inches  ;  then 


_/K    /.'    *8\ 

"5-1  'V      Ai/ 


and  A!  =  3 


The  values  of  the  modulus  of  wrenching/  are — • 

for  cast-iron        about        30000, 
for  wrought-iron     "  54000, 

and  taking  six  as  the  factor  of  safety,  if  we  put  the 
working  moment  of  torsion  in  the  formulae  instead 
of  the  wrenching  moment,  we  may  put  instead  of/ 

for  cast  iron          5000, 

for  wrought  iron  9000. 

Hence  we  get  for  w,  the  working  stress,  with 
solid  shafts, 

5000  A'      980  A3  , 

Wj=    -  .,  T    = 5 —  for  cast-iron    ...   (2.) 

0.1  I  I 

9000  A1      1765  A3- 
=    -t1  j    =  — j —  for  wrought-iron..(3.) 

On  this  principle  I  have  calculated  the  follow-1 
ing  tables  (pages  200,  201,)  giving  the  safe  mo- 

i 


200  MACHINERY  *OP   TRANSMISSION. 


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STRENGTH   AND   PROPORTIONS   OF   SHAFTS.    201 


^H  rH  i-H  (N  C^l  CM  CO 


OCil—  rHa 


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CC;'7-»?)-H"'CQ—hir 

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202  MACHINERY   OF  TRANSMISSION. 

ment  of  torsion  for  cylindrical  cast  and  wrought- 
iron  shafts,  and  also  the  working  stress  to  which 
they  may  be  subjected  at  the  circumference  of 
pullies  or  wheels  of  various  diameters.  In  cases 
where  the  horses'  power  transmitted  by  a  shaft 
is  given  instead  of  the  stress,  the  latter  may  be 
found  by  the  table  on  page  172. 

The  greatest  angle  of  torsion,  which  it  is  safe  to 
allow  in  a  line  of  shafting,  is  determined  by  the 
extension  of  the  material  within  the'elastic  limits. 
If  yaVu^h  °f  the  length  be  assumed  as  the  maxi- 
mum extension  with  the  safe  working  load,  then 
the  shaft  must  be  so  proportioned  that  the  angle 
of  torsion  is  less  than  that  given  by  the  following 
formula : 

_2284_L 
"  1000  d  '"{  ' 

where  L  is  the  length  of  the  shaft  in  feet,  d  its 
diameter  in  inches,  and  •*•  the  angle  of  torsion  in 
degrees. 

It  is  convenient  to  estimate  the  ultimate  resist- 
ance of  shafts  to  torsion,  not  only  as  a  statical 
pressure  acting  at  a  leverage,  but  also  in  horses' 
power.  Now  the  stress  resulting  from  the  trans- 
mission of  power  must  evidently  increase  in  pro- 
portion to  the  power,  and  decrease  in  proportion 
to  the  velocity.  A  shaft  will  transmit  100  horses' 
power  at  80  revolutions  a  minute  with  no  more 
stress  than  it  would  transmit  50  horses'  power  at 


i 


STRENGTH  AND   PROPORTIONS   OF   SHAFTS.    203 

40  revolutions,  or  25  horses'  power  at  20  revolu- 

TT 

tions.     Hence  the  torsion  varies  as  -,  where  H  is 

R 

the  number  of  horses'  power  per  minute,  and  K 
the  number  of  revolutions  per  minute. 

Buchanan's  rules  for  the  power  transmitted  by 
shafts  are:  —  * 

For  fly-wheel  shafts 


For  shafts  of  water-wheel  gearing  and  other 
heavy  work, 


For  shafts  of  ordinary  mill  gearing 


An    ordinary   allowance    for   wrought-iron 
shafting  in  practice  is 

5X250}  •  •  •  (5-) 
From  the  foregoing  observations  in  regard  to 
torsion,  and  the  power  of  transmission  of  shafts 
at  different  velocities,  it  is  a  desideratum  of  much 
importance  to  the  engineer,  so  to  proportion  shafts 
in  relation  to  their  lengths  as  well  as  velocities, 
as  to  be  within  the  limits  of  sensible  permanent 

*  These  rules  will  be  found  in  the  second  edition  of  Bu- 
chanan, at  pages  328,  et  seq, 


204  MACHINERY  OP  TRANSMISSION. 

torsion  and  flexure,*  and  at  the  same  time  to  in- 
crease the  speeds  in  a  given  ratio  to  the  velocities 
of  the  machine  and  the  nature  of  the  work  it  has 
to  execute.  In  the  above  disquisition  we  have 
only  given  the  law  and  the  safe  measure  of  tor- 
sion as  regards  length  and  area,  but  much  must 
still  depend  on  the  calculation  and  judgment  of 
the  millwright  and  engineer ;  in  its  application  to 
the  character  of  the  work  they  have  to  perform, 
and  the  resistances  they  have  to  overcome. 

From  formula  (5.)  the  following  table  (page  205) 
has  been  calculated,  giving  the  diameter  necessary 
to  transmit  from  1  to  150  horses'  power  at  from 
10  to  1000  revolutions  per  minute. 

4.   Velocity  of  Shafts, 

As  the  quality  of  the  material  employed  for 
the  construction  of  shafts  enters  largely  into  the 
calculation  of  their  strength,  so  also  the  velocity 
at  which  they  revolve  becomes  an  important 
element  in  the  calculation  of  the  work  transmitted 
by  them.  In  all  cases  where  machinery  has  to 
be  driven  at  a  high  speed,  it  is  advantageous  and 

*  Although  we  speak  of  the  limits  of  permanent  torsion, 
we  are  not  prepared  to  fix  these  limits,  as  we  find  that 
what  produces  a  permanent  set  in  any  material,  however 
minute  a  fraction  it  may  be,  will,  in  process  of  time,  if  con- 
tinned,  and  often  repeated,  lead  to  fracture.  This  law  ap- 
plies to  every  description  of  strain  or  material,  and  we 
may  therefore  consider  that  there  are  limits  to  endurance, 
however  distant  that  may  be. 


STRENGTH  AND   PROPORTIONS   OF   SHAFTS.    205 


«NCNe*6»webMWWW«-«i<^l' 


206  MACHINERY   OF   TRANSMISSION. 

even  essential  to  run  the  shafting  at  a  proportion- 
ate velocity.  If,  for  example,  there  are  a  series 
of  machines  running  at  five  hundred  revolutions 
per  minute,  it  will  be  advisable  to  run  the  shafts 
at  half  that  speed,  by  which  means  the  following 
very  important  advantages  will  be  gained. 

There  will  be  a  great  saving  in  the  weight  of 
the  shafts,  for  with  a  slow  motion  of  fifty  revolu- 
tions per  minute,  fully  three  times  the  weight 
would  be  necessary  to  transmit  the  same  power. 
There  would  also  be  a  saving  in  original  cost  in 
the  power  absorbed,  and  in  maintenance. 

Shafts  running  at  low  velocities  are  cumber- 
some, heavy,  and  expensive  to  repair.  They  are 
costly  in  the  first  instance,  and  they  block  up  the 
rooms  of  the  mill  with  large  drums  and  pullies, 
obstructing  the  light,  which,  in  factories,  is  a  con- 
sideration of  very  great  importance. 

At  the  commencement  of  the  present  century, 
mills  were  geared  with  ponderous  shafts,  such 
as  those  just  described.  They  were  generally 
of  cast-iron,  square,  and  badly  coupled,  and  the 
power  required  to  keep  them  in  motion  was  in 
some  cases  almost  equal  to  that  required  by  the 
machinery  they  had  to  drive.  In  the  present 
improved  system,  with  light  shafts  accurately 
fitted  and  running  at  high  velocities,  the  work 
which  previously  was  absorbed  in  transmission  is 
now  conveyed  to  the  machinery  of  the  mill. 

I  may  safely  ascribe  my  own  success  in  life 


STRENGTH  AND  PROPORTIONS   OF  SHAFTS.    207 

and  that  of  my  friend  and  late  partner,  Mr.  James 
Lillie,  to  the  saving  of  power  effected  by  increas- 
ing threefold  the  velocity  of  the  shafting  in  mills 
more  than  forty  years  ago.  The  introduction  of 
light  iron  shafting  not  only  enabled  the  manufac- 
turer to  effect  a  considerable  saving  in  the  origi- 
nal cost,  but  a  still  greater  saving  was  effected  in 
power,  whilst  it  relieved  the  mills  from  the  pon- 
derous wooden  drums  and  heavy  shafting  then  in 
use,  and  established  an  entirely  new  system  of 
operations  in  the  machinery  of  transmission. 

5.  Length  of  Journals.* 

Another  consideration  of  considerable  import- 
ance to  the  smooth  and  safe  working  of  shafting 
is  the  length  of  the  journals.  From  a  number  of 
years'  experience  I  have  been  led  to  believe,  that 
with  cast-iron,  one  and  a  half  times  the  diameter 
of  the  shaft  is  the  best  proportion  for  the  length 
of  the  bearing,  and  with  wrought  iron,  one  and 
three  quarters  the  diameter.  On  the  question  of 
shafts  revolving  in  the  steps  of  plummer  blocks 
and  the  proportions  necessary  to  effect  motion 
without  danger  of  heating,  it  is  essential  (without 
entering  largely  into  the  laws  of  friction  on  bodies 

*  Rules  for  the  diameters  of  gudgeons  or  journals  for 
those  cases  in  which  they  are  calculated  independently  of 
the  diameter  of  the  shaft,  are  given  in  Mills  and  Millwork, 
vol.  i.  p.  116. 


208 


MACHINERY   OF   TRANSMISSION. 


in  contact)  that  we  should  ascertain  from  actual 
practice  and  long-tried  experience  the  best  form 
of  journals  of  shafts  adapted  for  that  purpose. 
The  lengths  proportionate  to  the  diameters  have 
already  been  given,  but  we  have  yet  to  consider 
the  dimensions  of  the  journals  of  large  shafts 
where  they  are  small  in  comparison  with  the 
pressure  or  the  weight  they  have  to  sustain.  Let 
us,  for  example,  take  a  fly-wheel  shaft  and  the 
foot  or  toe  of  a  line  of  vertical  shaft  extending  to 
a  height  of  six  or  seven  stories  in  a  mill  filled 
with  machinery,  and  we  have  the  safe  working 
pressure  per  square  inch  as  indicated  in  the  last 
column  in  the  following  table : — 


t« 

g. 

£ 

"•o 

—  " 

*  s 

Description  of  Shaft. 

c<«.  rc 

p  S 

rj 

1^ 

.£|| 

S;£  J 

<S|   _ 

-=.5 

•?  g-2 

"S.  £.3 

—  £  « 

rf  a  a 

«c. 

»-- 

>   B.O 

Fly-wheel  shaft  wrought-iron  

18  X  H 

252 

45,024 

178-21 

—  y  11 

95 

23,061 

242-70 

Horizontal  shaft  cast-iron  

15  X  1° 

150 

6,000 

40-00 

6  X    3 

18 

540 

30-00 

Ditto        ditto        ditto       

2  X    * 

8 

160 

20-00 

From  the  above  it  will  be  seen  that -in  fly-wheel 
shafts  the  pressure  should  never  exceed  180  Ibs. 
per  square  inch,  and  in  that  of  the  toes  of  vertical 
shafts  240  Ibs.  per  square  inch.  Even  with  this 
latter  pressure  it  is  difficult  to  keep  the  shafts 
cool,  and  it  requires  the  greatest  possible  care  to 
keep  them  free  from  dust  or  any  minute  particles 


STRENGTH   AND   PROPORTIONS  OF  SHAFTS.    209 

of  sand  or  other  sharp  substances  getting  into  the 
steps.  The  feet  of  vertical  shafts  also  require  the 
very  best  quality  of  gun  metal  for  the  shaft  to 
run  in,  and  fine  limpid  oil  for  Imbrication  to  pre- 
vent the  toe  from  cutting.  It  is,  moreover,  neces- 
sary'for  the  shaft  to  fit  well  on  the  bottom  of  the 
step,  and  not  too  tight  on  the  sides,  and  to  have  a 
fine  polish. 

Another   point   for  Fig. us. 

consideration  is  the 
proper  form  of  the 
journals  of  shafts,  and 
that  is,  they  should  never  have  the  journal  turned 
or  cut  square  down  to  the  diameter,  but  hollowed 
in  the  form  shown  in  the  figure  at  a,  a  a  a. 
From  a  series  of  interesting  experiments  it  has 
been  shown  that  the  square-cut  shaft  loses  nearly 
one-fifth  of  its  strength,  and  by  simply  curving 
out  the  shaft  at  the  collars  in  the  form  described, 
the  resistance  to  strain  is  increased  one-fifth  or 
in  that  proportion. 

6.  Friction. 

On  the  subject  of  friction  much  cannot  be  said. 

We   may,  however,    adduce   a   few   experiments 

from  Morin  and  Riviere,  which  appear  to  bear  out 

our  previous  experience  of  the  length  of  journals. 

In  the  years  1831,  1832  and  1833,  a  very  ex- 

tensive  set  of  experiments  we're  made  at  Metz  by 

M.  Morin,  under  the  sanction  of  the  French  govern- 

19* 


210  MACHINERY   OF  TRANSMISSION. 

ment,  to  determine,  as  nearly  as  possible,  the  laws 
of  friction,  and  by  which  the  following  were  fully 
established: — 

When  no  unguent  is  interposed,  the  friction  of 
any  two  surfaces,  whether  of  quiescence  or  of 
motion,  is  directly  proportional  to  the  force  with 
which  they  are  pressed  perpendicularly  together ; 
so  that  for  any  two  given  surfaces  of  contact  there 
is  a  constant  ratio  of  the  friction  to  the  perpen- 
dicular pressure  of  the  one  surface  upon  the  other. 
Whilst  this  ratio  is  thus  the  same  for  the  same  sur- 
faces of  contact,  it  is  different  for  different  surfaces 
of  contact.  The  particular  value  of  it  in  respect 
to  any  two  given  surfaces  of  contact,  is  called  the 
coefficient  of  friction  in  respect  to  those  surfaces. 

When  no  unguent  is  interposed,  the  amount  of 
the  friction  is,  in  every  case,  wholly  independent 
of  the  extent  of  the  surfaces  of  contact ;  so  that 
the  force  with  which  two  surfaces  are  pressed  to- 
gether, being  the  same,  their  friction  is  the  same, 
whatever  be  the  extent  of  their  surfaces  of  contact. 

That  the  friction  of  motion  is  wholly  indepen- 
dent of  the  velocity  of  the  motion. 

That  where  unguents  are  interposed,  the  coeffi- 
cient of  friction  depends  upon  the  nature  of  the 
unguent,  and  upon  the  greater  or  less  abundance 
of  the  supply.  In  respect  to  the  supply  of  the 
unguent,  there  are  two  extreme  cases, — that  in 
which  the  surfaces  of  contact  are  but  slightly 
rubbed  with  the  unctuous  matter,  as,  for  instance, 


STRENGTH   AND   PROPORTIONS   OF   SHAFTS.    211 

with  an  oiled  or  greasy  cloth,  and  that  in  which  a 
continuous  stratum  of  unguent  remains  continually 
interposed  between  the  moving  surfaces;  and  in 
this  state  the  amount  of  friction  is  found  to  be  de- 
pendent rather  upon  the  nature  of  the  unguent 
than  upon  that  of  the  surfaces  of  contact.  M. 
Morin  found  that  with  unguents  (hog's  lard  and 
olive  oil)  interposed  in  a  continuous  stratum  be- 
tween surfaces  of  wood  on  metal,  wood  on  wood, 
and  metal  on  metal,  when  in  motion,  have  all  of 
them  very  nearly  the  same  coefficient  of  friction, 
being  in  all  cases  included  between  '07  and  €08. 
The  coefficient  for  the  unguent  tallow  is  the  same, 
except  in  that  of  metals  upon  metals.  This  un- 
guent appears  to  be  less  suited  for  metallic  surfaces 
than  the  others,  and  gives  for  the  mean  value  of 
its  coefficient  under  the  same  circumstances  "10. 
Hence  it  is  evident  that  where  the  extent  of  the 
surface  sustaining  a  given  pressure  is  so  great  as 
to  make  the  pressure  less  than  that  which  corres- 
ponds to  a  state  of  perfect  separation,  this  greater 
extent  of  surface  tends  to  increase  the  friction  by 
reason  of  that  adhesiveness  of  the  unguent,  depen- 
dent upon  its  greater  or  less  viscosity,  whose 
effect  is  proportional  to  the  extent  of  the  surfaces 
between  which  it  is  interposed. 

Mr.  G.  Eennie  found,  from  a  mean  of  experi- 
ments with  different  unguents  on  axles  in  motion, 
and  under  different  pressures,  that  with  the  un- 
guent tallow,  under  a  pressure  of  from  1  to  5  cwt., 


212 


MACHINERY   OF   TRANSMISSION. 


the  friction  did  not  exceed  ^th  of  the  whole  pres- 
sure ;  when  soft  soap  was  applied  it  became  g\th ; 
and  with  the  softer  unguents  applied,  such  as  oil, 
hog's  lard,  etc.,  the  ratio  of  the  friction  to  the 
pressure  increased ;  but  with  the  harder  unguents, 
as  soft  soap,  tallow,  and  anti-attrition  composition, 
the  friction  considerably  diminished ;  consequently 
to  secure  effective  lubrication,  the  nature  of  the 
unguent  must  be  accommodated  to  the  pressure 
or  weight  tending  to  force  the  surfaces  together. 

TABLE  OF  COEFFICIENTS  OF  FRICTION  UNDER  PRESSURES 
INCREASED  CONTINUALLY  UP  TO  LlMITS  OF  ABRASION. 
BY  MR.  G.  RENNIE. 


Coefficients  of  Friction. 

Pressures  per 
square  inch. 

Wrought-iron  upon 

Wrought-iron 

Steel  upon 

Brass  upon 

wrought-irou. 

upon  cast-iron. 

cast-iron. 

cast-iron. 

32-5    Ibs. 

•140 

•174 

•166 

•157 

1-66  cwts. 

•250 

•275 

•300 

•225 

2-00 

•271 

•292 

•333 

•219 

2-33 

•285 

•321 

•340 

•214 

2-66 

•297 

•329 

•344 

•211 

300 

•312 

•333 

•347 

•215 

3-33 

•350 

•351 

•351 

•206 

3-66 

•376 

•353 

•353 

•205 

4-00 

•395 

•365 

•354 

•208 

4-33 

•403 

•366 

•356 

•221 

4-66 

•409 

•366 

•357 

•223 

5-00 

— 

•367 

•358 

•233 

5-33 

— 

•367 

•359 

•234 

5-66 

— 

•367 

•367 

•235 

6-00 

— 

•376 

•403 

•233 

6-33 

— 

•434 

— 

•234 

6-66 

— 

— 

— 

•235 

7-00 

— 

— 

— 

•232 

7-33 

— 

— 

— 

•273 

STRENGTH   AND  PROPORTIONS   OF   SHAFTS.    213 

From  a  paper  lately  read  at  the  Institution  of 
Civil  Engineers  in  London,  on  the  comparative 
friction  of  steam  engines  of  different  modifications, 
it  appears  that,  as  respects  the  friction  caused  by 
the  strain,  if  the  beam  engine  be  taken  as  the 
standard  of  comparison — 

The  vibrating  engine has  a  gain  of  1-1  per  cent. 

The  direct  engine  with  slides ....  "  loss  of  1/8  " 

Ditto  with  rollers "  gain  of  0-8  " 

Ditto  with  a  parallel  motion  "  gain  of  1/3  " 

It  also  states,  as  an  opinion,  that  excessive  al- 
lowance for  friction  has  hitherto  been  made  in 
calculating  the  effective  power  of  engines  in  gen- 
eral; as  it  is  found  practically  by  experiments 
with  the  engines  at  the  Blackwall  Eailway,  and 
also  with  other  engines,  that  where  the  pressure 
upon  the  piston  is  about  12  Ibs.  per  square  inch, 
the  friction  does  not  amount  to  more  than  If  Ibs.; 
and  also  that  by  experiments  with  an  indicator  on 
an  engine  of  50  horse-power,  at  Truman,  Hanbury 
and  Co.'s  brewery,  the  whole  amount  of  friction 
did  not  exceed  5  horse-power,  or  -j^th  of  the 
whole  power  of  the  engine. 

7.  Lubrication. 

On  this  question  it  is  necessary  to  observe  that 
the  durability  of  shafts,  and  their  easy  working, 
depends  on  the  way  in  which  they  are  lubricated, 
and  the  description  of  unguent  used  for  that  pur- 


214  MACHINERY   OF   TRANSMISSION. 

pose.  We  have  already  seen  the  difference  which 
exists  in  the  coefficient  of  friction  from  the  use  of 
different  kinds  of  unguents,  and  \re  have  now  to 
consider  what  system  of  lubrication  should  be 
adopted  to  lessen  the  friction  and  maintain  smooth 
surfaces  on  the  journals  of  shafts.  In  large  cot- 
ton mills  I  have  known  as  much  as  ten  to  fifteen 
horses'  power  absorbed  by  a  change  in  the  quality 
of  the  oil  used  for  lubrication;  and  in  cold 
weather,  or  when  the  temperature  of  the  mill  is 
much  reduced  (as  is  generally  the  case  when 
standing  over  Sunday),  the  power  required  on  a 
Monday  morning  is  invariably  greater  than  at 
any  other  time  during  the  week. 

It  is,  therefore,  necessary  in  most  mills — parti- 
cularly those  employed  in  textile  manufacture — 
to  retain  a  uniform  temperature,  and  to  employ 
the  best  quality  of  oil  for  lubricating  the  ma- 
chinery, as  well  as  the  shafts  of  the  mill. 

The  best  lubricators  are  pure  sperm  and  olive 
oils ;  they  should  be  clean  and  limpid,  and  spar- 
ingly applied,  as  it  is  a  profligate  waste  of  valua- 
ble material  to  pour,  as  is  not  unfrequently  done, 
large  quantities  of  oil  on  the  bearings,  nine-tenths 
of  which  run  on  to  the  floor,  and  cover  the  shafts 
and  hangers  with  a  coat  of  glutinous  matter,  that 
soon  hardens,  and  accumulates  nothing  but  filth. 

This  process  of  oiling  shafts  is  generally  left  to 
the  most  negligent  and  most  untidy  person  in  the 
establishment ;  and  the  result  is,  that  every  open- 


STRENGTH  AND   PROPORTIONS   OF  SHAFTS.    215 


Fig.  117. 


ing  for  the  oil  to  get  to  the  bearings  is  plugged 
up,  the  brass  steps  are  cut  by  abrasion,  and  the 
necks  or  journals  of  the  shafts  destroyed.  In  the 
best  regulated  establishments  this  is  certainly  not 
the  case,  as  the  greatest  possible  care  is  observed 
in  selecting  the  best  kinds  of  oil,  and  that  used 
with  attention  to  cleanliness  and  strict  economy  in 
its  application. 

To  save  power  and  effect  economy  in  the  use  of 
lubricants,  several  schemes  have  been  adopted  for 
attaining  a  continuous  system  of  lubrication. 
None  of  them  appears  to  answer  so  well  as  that 
which  consists  of  a  small  cistern,  a,  fig.  117, 
which  contains  a  quantity 
of  oil,  and  is  fixed  on  the 
top  of  the  plummer  block. 
In  the  centre  of  the  cistern 
is  a  tube,  which  stands  a 
little  above  the  level  of  the  oil ;  and  into  this  is 
inserted  a  woollen  thread,  with  its  end  descending 
a  short  distance  below  the  surface  of  the  oil  in 
the  cistern ;  and  when  properly  saturated,  the  oil 
rises  by  capillary  attraction,  and  flows  gently,  in 
very  minute  quantities,  on  to  the  neck  of  the 
shaft.  From  this  description  it  will  be  seen  that 
the  quantity  used  can  be  regulated  to  the  greatest 
nicety,  and  sufficient  to  lubricate  the  bearings 
without  waste.  Other  plans  have  been  devised 
for  the  same  object,  but  none  of  them  seems  to 
answer  so  well  as  that  just  described. 


216  MACHINERY   OF  TRANSMISSION. 


CHAPTER  IV. 

ON  COUPLINGS  FOR  SHAFTS  AND  ENGAGING  AND  DISENGAGING 
GEAR. 

IN  every  description  of  mill  where  the  ma- 
chinery is  spread  over  a  large  area,  and  at  a  dis- 
tance from  the  moving  power,  it  is  necessary  to 
have  long  lines  of  shafting,  revolving  at  the  re- 
quired velocity.  Such  lines  are  seldom  made  in 
one  piece ;  short  lengths  must,  therefore,  be  coup- 
led together,  so  as  to  form  an  unbroken  line,  ex- 
tending, in  most  cases,  the  whole  length  of  the  mill. 

When  cast  iron  shafts  were  substituted  for 
wood,  a  square  coupling-box,  made  in  one  piece, 
was  generally  used,  so  as  to  slide  over  the  two 
ends  of  the  shafts,  or  in  two  pieces,  bolted  to- 
gether, as  shown  in  figs.  118  and  119. 

Fig.  118. 


In  the  former  case  the  box  was  slipped  on 
loose,  and  the  adjustment  was  so  imperfect  that 
the  shafts  rose  and  fell  in  the  box  at  every  revolu- 
tion, destroying  gradually  any  accuracy  of  fitting 
which,  in  the  first  instance,  had  been  attained. 


ON   COUPLINGS. 


217 


After  the  square-box  coupling  came  the  claw, 
or  two-pronged  coupling,   made   in    two    parts, 


Fig.  119. 


218 


MACHINERY   OF   TRANSMISSION. 


wedged,  but  more  frequently  keyed  on  to  the  ends 
of  the  shafts,  as  shown  in  ^fig.  120.     This  was  a 
great  improvement,  as  the  leverage  of  the  bear- 
Fig.  122. 


Fig.  123. 


ing  parts-  was  greatly  increased,  and  the  coupling, 
in  consequence,  became  more  durable.  i 

A  description  of  half-lap  coupling  was  intro- 
duced by  the  late  Mr.  Hewes.     It  was  formed  by 


ox  COUPLINGS.  219 

the  lapping  over  a  part  of  the  end  of  each  shaft, 
which  was  cast  square.  A  square  box  was  also 
fitted  over  the  two  ends,  so  as  to  bind  them  to- 
gether, and  three  keys  were  inserted  on  the  top 
side,  as  shown  in  fig.  121.  The  objections  to  this 
coupling  were  the  difficulty  of  fitting  and  the 
loosening  of  the  keys,  which  made  a  creaking 
noise  with  every  revolution  of  the  shaft. 

Another  coupling,  still  in  use,  is  the  disc.  It 
consists  of  two  discs  or  flanches,  one  on  the  end 
of  each  shaft,  bolted  together  by  four  bolts,  as 
shown  in  fig.  122.  This  coupling  was  superior  to 
all  the  preceding,  when  properly  bored  and 
turned,  so  as  to  have  its  faces  accurately  perpen- 
dicular to  the  shafting. 

The  best  coupling  for  general  purposes,  and 
the  most  accurate  and  durable,  is  the  circular 
half-lap  coupling,  introduced  into  my  own  works 
nearly  forty  years  ago.  It  is  perfectly  round,  and 
consists  of  two  laps,  turned  to  a  gauge,  and,  when 
put  together  by  a  cutting  machine,  it  forms  a 
complete  cylinder,  as  shown  in  fig.  123.  A  cylin- 
drical box  is  fitted  over  these,  and  fixed  by  a  key, 
grooved  half  into  the  box  and  half  into  the  shaft. 
The  whole  is  then  turned  in  the  lathe  to  the  same 
centres  as  the  bearings  of  the  shaft,  and  by  this 
process  a  degree  of  accuracy  is  attained  which 
cannot  be  surpassed,  nor  is  any  other  coupling  so 
neat  and  so  well  adapted  for  the  transmission  of 
power. 


220 


MACHINERY   OF  TRANSMISSION. 


The  proportions  of  this  coupling  are  found  by 
experiment  to  be — 

Twice  the  area  of  the  shaft  is  the  area  of  the 
coupling. 

The  length  of  the  lap  is  the  diameter  of  the 
shaft. 

And  the  length  of  the  box  is  twice  the  diameter 
of  the  shaft. 

These  proportions  have  been  found  in  practice 
to  answer  every  purpose,  both  as  regards  strength 
and  the  wear  and  tear  of  the  joints. 

There  is  another  coupling  which  has  come  of 
late  years  extensively  into  use,  namely,  the  cylin- 

Pig.  124. 


drical  coupling,  with  butt  ends.  It  has  the  same 
proportions  as  the  former,  but  not  so  strong  nor 
so  durable  as  the  half-lap  coupling  of  the  same 
dimensions,  as  the  entire  force  of  torsion  is  trans 
mitted  through  the  key;  but  in  cases  whera 
strength  is  not  the  chief  object,  it  forms  a  chert  i 
and  effective  coupling. 


ENGAGING   AND   DISENGAGING   GEAR.        221 

8.  Disengaging  and  re-engaging  Gear. 

This  is  an  important  branch  of  mill-work,  re- 
quiring careful  consideration,  and  the  utmost 
exactitude  of  construction  when  ponderous  ma- 
chinery has  to  be  started,  without  endangering  the 
shafts  and  wheels.  This  is  most  strikingly  exem- 
plified in  the  case  of  powder  mills,  where  trains 
of  edge  stones  are  employed  for  grinding  the  gun- 
powder, and  in  rolling  and  calendering  machinery 
which  requires  well  fitted  friction-clutches  to  com- 
municate the  motion  by  a  slow  and  progressive 
acceleration  from  a  state  of  rest  to  the  required 
velocity. 

It  used  to  be  customary  in  cotton  and  silk  mills 
to  place  disengaging  clutches  at  the  point  of  con- 
nection of  the  upright  or  driving  shaft  and  the 
main  shafting  in  each  room,  so  that,  in  case  of  ac- 
cident, a  room  full  of  machinery  could  be  thrown 
out  of  gear  at  once.  But  these  provisions  were 
found  unsteady  in  practice,  and  rather  tended  to 
increase  than  to  diminish  the  number  of  accidents, 
owing  chiefly  to  the  time  lost  in  disengaging,  and 
the  breakages  which  occurred  in  attempting  to 
place  the  machinery  in  gear  again,  when  the  en- 
gine was  running  at  full  speed.  It  has,  conse- 
quently, been  found  safer  to  have  a  permanent 
connection  between  the  main  lines  of  shafting 
throughout  the  mill,  and  signals  from  each  room 
into  the  engine-house,  in  case  of  accident. 

When  the  construction  of  mill  gearing  was  less 
19* 


222 


MACHINERY   OF   TRANSMISSION. 


perfect  than  it  is  at  present,  the  main  shaft  driving 
the  machinery  in  a  room  was  thrown  out  of  gear 
by  a  lever,  which  contained  the  steps,  and  sup- 
ported the  end  of  the  horizontal  shaft  and  wheel, 

Fig.  126. 


which  geared  into  that  on  the  upright  shaft,  as 
shown  in  fig.  125,  with  a  rope  at  the  end  of  the 
lever  o  to  pull  it  out  of  gear.  This  mode  of  dis 
engaging  wheels  was  very  ineffective,  as  in  many 
mills  there  are  three  bevel  wheels  gearing  into 
that  on  the  upright  b,  and  it  becomes  complicated 


ENGAGING   AND   DISENGAGING   GEAR.       223 

and  dangerous  to  have  movable  levers  to  each. 
To  remedy  these  defects,  standards  or  plummer- 
blocks,  with  a  movable  slide  e,  fig.  126,  in  which 
the  end  of  the  shaft  revolved,  was  introduced.  To 
the  top  of  this  slide  was  attached  a  lever  a,  with  a 
handle  b,  by  which  it  could  be  drawn  out  of  gear ; 
and  the  link  c,  falling  along  with  the  lever,  re- 

Jig.  126.  i 


tained  the  shaft  out  of  gear  until  the  mill  was 
stopped. 

All  these  contrivances  were,  however,  found  in- 
operative on  a  large  scale,  as  the  shafts  and  wheels 
got  out  of  order;  and  it  was  ultimately  found- 
essentially  necessary  to  make  them  stationary,  by 


224  MACHINERY   OF   TRANSMISSION. 

screwing  the  plummer-block  down  to  the  frame 
which  connects  the  shafts  and  wheels. 

Several  devices  have  been  employed  for  the 
purpose  of  rapidly  engaging  and  disengaging  ma- 
chines from  the  driving  shaft.  The  best  of  all  are 

fig.  12T. 


the  fast  and  loose  pulleys,  with  a  travelling  strap. 
Thus,  in  fig.  127  a  is  the  driving  shaft,  acting 
upon  two  pulleys  e  and  d,  fixed  on  the  driving 
spindle  of  the  machine  b ;  one  of  them,  d,  is  keyed 
fast,  and  the  other  runs  loose.  When  the  machine 
is  at  work  the  strap  is  on  the  fast  pulley  (/,  and 


ENGAGING   AND   DISENGAGING   GEAR.       225 

when  it  is  necessary  to  stop,  it  is  moved  by  a 
forked  lever  on  to  the  loose  pulley  e,  which  re- 
volves with  the  strap  without  acting  on  the  ma- 
chine. The  machine  is  thrown  into  gear  with 
equal  ease  by  moving  the  strap  on  to  the  fast  pul- 
ley d.  Once  on  either  of  the  pulleys,  the  strap  is 
held  in  position  without  any  danger  of  moving  by 
the  slight  curvature  of  the  pulley,  as  already  ex- 
plained. The  forked  lever  must  act  on  that  side 
of  the  strap  which  runs  toward  the  pulleys,  and 
not  on  that  which  leaves  them. 

A  second  and  equally  effective  process  for  start- 
rig.  128. 


226 


MACHINERY   OF   TRANSMISSION. 


ing  or  stopping  machinery  is  shown  in  fig.  128. 
A  leather  strap  is  hung  loosely  over  the  driving 
and  driven  pulleys  a  and  b,  so  that,  left  to  itself, 
the  friction  is  not  sufficient  to  communicate  mo- 
tion to  the  pulley  on  the  shaft  b ;  but  a  tightening 
pulley  fixed  on  a  suitable  lever  e  is  forced  against 
it  by  pulling  the  rope  c,  which  bends  the  strap 
tightly  upon  the  pulley  b,  and  gives  motion  to  the 
machine.  This  arrangement  is  in  general  use  for 
sack  teagles  in  corn  mills,  and  for  some  other  pur- 
poses. The  same  effect  is  sometimes  produced  by 
the  sack  teagles  being  fixed  on  the  lever,  and,  by 
raising  one  end,  the  strap  is  tightened,  and  the 
barrel  which  raises  the  load  is  caused  to  revolve. 


The  clutch  most  in  use  for  throwing  into  gear 
heavy  calendering   machines   is  a   clip   friction 


ENGAGING  AND  DISENGAGING   GEAR.        227 

hoop,  which  consists  of  a  sliding  box  a,  with  two 
projecting  horns  on  the  driving  shaft  b.  These 
horns,  when  slid  forward  by  a  lever  g,  working 
in  the  groove  c,  come  in  contact  with  the  friction 
hoop  d,  which  embraces  a  groove  in  a  second  box, 
keyed  upon  the  shaft  of  the  machine.  The  instant 
the  machine  receives  the  shock  of  engagement,  the 
clip  d  slides  in  its  groove,  until  the  friction  over- 
comes the  resistance,  and  the  callender  attains  the 
speed  of  the  driving  shaft.  The  object  of  the  fric- 
tion clip  is  to  reduce  the  shock  of  throwing  the 
clutch  suddenly  into  gear,  as  without  this  precau- 
tion any  attempt  to  move  instantaneously  a  pow- 
erful machine  from  a  state  of  rest  to  a  state  of 
motion  would  break  it  in  pieces. 

Friction  cones  are  also  much  used  for  this  pur- 
pose, and  when  carefully  executed  with  the  proper 
angle  are  safer  than  the  clutch  just  described.  The 
objection  to  the  friction  clutch  is,  that  the  whole 
driving  power  is  thrown  on  the  clip  at  once ; 
whereas,  with  the  cones,  the  parts  can  be  brought 
into  contact  with  the  greatest  nicety,  and  the  fric- 
tion regularly  increased  to  any  degree  of  pres- 
sure. Fig.  130  shows  this  description  of  disen- 
gaging gear ;  a  is  the  male  sliding  cone,  worked 
by  a  lever  in  the  usual  way,  b  the  female  cone, 
keyed  on  the  driven  shaft,  and  the  two  surfaces, 
when  brought  into  contact,  communicate  the  re- 
quired motion  with  perfect  safely. 

Machines  driven  by  friction,  and  requiring  to 


228  MACHINERY   OF  TRANSMISSION. 

Fig.  130. 
1 


be  frequently  stopped,  are  very  numerous.  Some 
of  the  lighter  description  are  driven  by  a  vertical 
shaft,  b,  fig.  131,  supporting  a  horizontal  disc, 

which  communicates 
motion  to  the  wheel 
a,  rolling  on  its  sur- 
face, and  gives  the 


Fig.  131. 


necessary  motion  to 
the  machine.  The 
advantage  of  this  fric- 
tion-wheel is,  that  the  velocity  of  the  machine  may 
be  increased  or  diminished  at  pleasure  by  moving 
the  wheel  a  nearer  to  or  farther  from  the  edge  of 
the  disc. 


ENGAGING  AND   DISENGAGING  GEAR.        229 

Fig.  132  is  another  combination  of  discs  suita- 
ble for  couplings  with  only  one  bearing.  The 
disc  b  is  keyed  on  one  shaft,  and  is  recessed  on 
the  face,  to  receive  the  smaller  disc,  c ;  this  disc  is 
sunk  flush  with,  the  face  of  the  other,  and  is 
screwed  tightly  up  to  it  by  means  of  the  ring  a, 

Tig.  132. 


•which  is  bolted  to  the  disc  b,  and  secures  that 
marked  c.  Between  the  three  plates,  a,  b  and  c, 
annular  pieces  of  leather  are  interposed,  which 
bring  them  all  to  a  proper  bearing. 

This  combination,  termed  a  friction  coupling,  is 

useful  for  preventing  breakage  of  the  connections 

1  in  case  of  a  sudden  stoppage  or  reversal  of  the 

,  motion.     It  is  plain  that  the  holding  power  of  the 

coupling  depends  simply  upon  the  lightness  with 

20 


230 


MACHINERY   OF   TRANSMISSION. 


which  the  discs  are  screwed  together,  and  the 
consequent  frictional  force  of  the  surfaces  of 
leather  and  metal. 

Besides  these  more  permanent  forms  of  coup- 
lings, there  are  other  contrivances  adopted  when 
the  object  to  be  attained  is  the  engagement  and 
disengagement  of  certain  parts  of  the  machinery 
or  gearing  during  the  course  of  operations. 

With  the  same  view  of  admitting  of  this  dis- 
engagement of  the  connection,  in  cases  of  sudden 

Fig.  133. 


stoppage  or  reversal,  the  coupling,  fig.   133,  is 
sometimes  employed. 

In   this  instance,  the  shaft  is  supposed   to  be 
continuous,  and  the  coupling   may  be  termed  a 


ENGAGING   AND  DISENGAGING   GEAR.        231 

disengaging  coupling,  a  and  b  are  the  two  parts 
of  the  coupling,  formed  on  the  acting  faces  into 
alternate  projections  and  recesses,  such  that  they 
correspond  with  and  exactly  fit  into  each  other 
when  in  gear.  The  part  a  is,  in  this  example, 
cast  on  a  spur  or  bevel  wheel,  from  which  the 
motion  of  the  shaft  is  supposed  to  be  taken  off. 
Both  of  the  parts  a  and  b  are,  to  a  certain  extent, 
loose  on  the  shaft ;  the  former  being  capable  of 
moving  round  on  it, '.though  deprived  of  longitu- 
dinal motion  by  washers  and  a  collar,  marked  e, 
and  the  latter  being  free  to  slide  on  the  shaft, 
though  prevented  from  turning  on  it  by  a  sunk 
key,  which  slides  in  a  slot  inside  the  clutch  or 
sliding  piece  b.  The  mechanism  is  put  into  gear 
by  means  of  the  lever  d,  which  terminates  in  a 
fork  with  cylindrical  extremities  c;  and  it  is 
obvious  that,  by  the  contact  of  the  flat  faces  of  a 
and  b,  the  latter  will  immediately  carry  with  it 
the  other  part  at  the  same  speed  as  the  shaft. 
Supposing,  now,  that  the  motion  of  the  wheel  a 
is  suddenly  accelerated,  the  oblique  faces  of  the 
couplings  immediately  fall  out  of  contact,  and 
slide  free  of  each  other,  leaving  the  couplings 
clear,  and  the  shaft  free  to  continue  in  motion. 

In  the  old  form  of  this  contrivance,  known  as 
the  sliding  bayonet  clutch,  the  part  b,  instead  of 
the  toothlike  projections  on  the  face,  had  two  or 
more  prongs  which  laid  hold  of  corresponding 
snugs  cast  on  the  face  of  the  part  a,  which,  more 


232  MACHINERY  OF   TRANSMISSION. 

over,  was  usually  a  broad  belt  pulley,  introduced 
with  a  view  to  modify  the  shock  on  the  gearing 
on  throwing  the  clutch  into  action. 

In  an  older  form  still,  the  pulley  was  made  to 
slide  end  long  on  the  shaft.  A  form  analogous  to 
this  was  known  as  the  "lock  pulley,"  a  few  speci- 
mens of  which  still  remain  in  the  older  factories. 
Instead  of  the  end  long  motion  common  to  the 
other  modes,  the  parts  were  "  locked  "  together  by 
a  bolt  fixed  upon  the  side  of  the  pulley,  and 
which,  when  shifted  toward  the  axis,  engaged 
with  an  arm  of  a  cross,  of  which  the  part  b,  in 
the  preceding  figure,  is  the  modern  representative. 
The  bolt  was  wrought  by  means  of  a  key  and 
stop,  the  turning  of  the  key  throwing  back  the 
bolt,  and  thereby  unlocking  and  disengaging  the 
pulley.  The  form  of  coupling  represented  by  fig. 
133  is  particularly  applicable  when  the  impelling 
power  is  derived  from  two  sources — a  circum- 
stance which  frequently  occurs  in  localities  afford- 
ing water  power  to  some  extent,  and  yet  not  in 
sufficient  abundance  for  the  demands  of  the  work. 
The  deficiency  is  usually  supplied  by  a  steam-en- 
gine ;  and  the  two  powers  are  concentrated  in  the 
main  line  of  shafting  by  a  coupling  of  the  kind 
depicted.  In  cases  of  this  kind,  the  speed  of  the 
shafting  being  fixed,  and  the  supply  of  water  in- 
constant, the  power  of  the  water-wheel  ought  to 
be  thrown  upon  the  wheel  a  a,  and  that  of  the 
engine  upon  the  shaft  at  another  point.  By  this 

I 


ENGAGING  AND   DISENGAGING   GEAR.        233 

arrangement,  the  speed  of  the  line  can  be  exactly 
regulated  by  working  the  engine  to  a  greater  or 
less  power,  according  to  the  supply  of  water. 
The  proper  speed  of  the  water-wheel  will  like- 
wise be  maintained,  which  is  of  importance  in 
economising  the  water  power. 

"  The  same  form  of  coupling  is  also  used  occa- 
sionally for  engaging  and  disengaging  portions  of 
the  machinery.  But  for  this  purpose  the  object 
is  to  obtain  a  mode  of  connection  by  which  the 
motion  may  be  commenced  without  shock;  for, 
in  consequence  of  the  inertia  of  all  material 
things — that  is,  the  tendency  which  every  portion 
of  matter  has,  when  at  rest,  to  remain  at  rest,  and 
when  in  motion,  to  continue  to  move — the  parts 
of  the  mechanism,  when  acted  upon  too  suddenly 
by  a  moving  power,  are  liable  to  fracture  and  dis- 
arrangement. It  is  a  law  in  mechanics  that  when 
a  body  is  struck  by  another  in  motion  some  time 
elapses  before  it  is  diffused  from  the  point  struck 
through  the  other  parts;  consequently,  if  the 
parts  receiving  the  blow  have  not  sufficient  elasti- 
city and  cohesive  force  to  absorb  the  whole  mo- 
mentum of  the  striking  body  till  the  motion  be 
transmitted  to  the  centre  of  rotation,  fracture  of 
the  body  struck  must  necessarily  ensue.  Hence, 
in  a  system  of  mechanism,  any  parts  intended  to 
be  acted  upon  suddenly  by  others  in  full  motion 
ought  not  only  to  be  strong,  but  they  ought  to  be 
capable  of  yielding  on  the  first  impulse  of  the 
20* 


234  MACHINERY   OF    TRANSMISSION. 

impelling  force  with  as  little  resistance  as  possible, 
and  gradually  bri»g  the  whole  weight  into  motion. 
The  common  mode  of  driving  by  belts  and  pul- 
leys accomplishes  this  object  very  satisfactorily. 
In  this  the  elasticity  of  the  belt  comes  into  action  ; 
and  being  thrown  upon  the  pulley  by  the  strap 
guide  or  fork,  it  continues  to  slip,  till,  by  the  fric- 
tion between  the  sliding  surfaces,  the  belt  grad- 
ually brings  the  quiescent  pulley  into  full  motion. 
This  mode  of  connection  is  unexceptional  when 
the  power  to  be  transferred  is  not  great ;  but  its 
application  to  large  machinery  is  attended  with 
inconvenience."* 

In  figs.  134  135,  two  other  forms  of  clutches 
are  shown,  as  often  used  to  connect  the  shafting 
of  different  parts  of  the  same  mill,  where  it  is  not 
necessary  to  throw  into  or  out  of  gear  when  run- 
ning at  full  speed.  They  consist  of  a  fixed  and 
sliding  box,  one  on  each  shaft,  with  teeth  or  pro- 
jections which  fit  in  corresponding  notches.  The 
sliding  box  has  a  groove  turned  in  it,  in  which  a 
forked  lever  works,  as  at  a,  fig.  134,  and  at  a,  fig. 
135,  by  which  it  is  drawn  backward  or  forward 
as  the  case  may  be.  The  peculiarity  of  the 
clutch,  fig.  135,  is  that  of  the  driving  shaft,  which, 
reversed  by  any  accident  in  its  motion,  as  is  not 
unfrequently  the  case  in  starting  and  stopping 
the  steam  engine,  the  sliding  clutch  is  forced  back 

*  Extract    from    Engineer's    and    Machinist's  Assistant, 
p.  144. 


ENGAGING   AND    DISENGAGING    GEAR.        235 


by  the  wedge-shaped  faces  of  the  projections,  and 
the  machinery  thrown  out  of  gear. 


Fig.  134. 


Fig.  13*). 


Fig.  136  shows  one  of  these  clutches  on  a  small 

scale,  fixed  on  a  line 
of  shafting  beneath 
the  floor  of  a  mill. 
It  is  placed  between 
two  standards  a  a, 
supporting  the  ends 
of  the  shaft,  and  the 
lever  b  working  on 
a  pivot  at  bottom, 
and  having  a  pin 
working  in  the 
groove  of  the  sliding 
clutch  box,  serves 


236          MACHINERY  OF  TRANSMISSION. 

for  throwing  the  driven  shaft  into  or  out  of  gear 
whenever  it  may  be  necessary. 

Another  ingenious  contrivance,  I  believe  in- 
vented by  Mr.  Bodmer,  is  shown  in  figs.  137  and 
138.  It  consists  of  a  box  a  a  running  loosely  on 
the  driving  shaft  s  s,  but  carrying  the  bevel  wheel 
b  b,  which  gears  into  another  wheel  on  the  driven 
shaft,  not  shown  in  the  figures.  Tightly  keyed 

Fig.137 . 


on  the  driving  shaft  s  s  is  a,  boss  c  c,  with  two 
trunnions,  on  which  slide  two  friction  sectors  k  k; 
the  outer  surface  is  coated  with  a  copper  plate, 


ENGAGING  AND   DISENGAGING    GEAR.       237 

accurately  fitting  the  interior  surface  of  the  run- 
ning box  a  a.  The  boss  c  c  carries  also  four  pro- 
jections e  e  e  e,  which  serve  as  guides  for  four 
screws,  alternately  left  and  right  handed,  and  at- 
tached to  the  nuts  //  and  levers  g  g ;  these  screws 
act  on  the  extremities  of  the  friction  slides  k  Jc,  so 
that  when  the  levers  g  g  are  drawn  back  they  are 
both  with  equal  pressure  forced  upon  the  inner 

Fig.  138. 


surface  of  the  box  a  a.  As  the  pressure  can  be 
very  regularly  and  gradually  brought  on  this  box 
through  the  levers  and  screws,  the  motion  of  the 
driving  shaft  s  s  is  communicated  with  perfect 


238  MACHINERY   OF  TRANSMISSION. 

regularity,  and  without  shock  to  the  bevel  wheel 
bb. 

In  the  above  description  I  have  given  such  ex- 
amples of  engaging  and  disengaging  gear  as  are 
most  commonly  in  use.  Others  of  a  more  com- 
plicated character  might  be  cited,  but  they  are 
not  to  be  recommended  as  applicable  in  general 
practice.  The  last  form,  figs.  137  and  138,  is. 
however,  specially  noticed  as  suitable  for  gun- 
powder mills,  where  the  greatest  possible  freedom 
from  shocks  is  essentially  necessary. 

9.  Hangers,  Plummer-bloc/cs,  etc.,  for  carrying 

Shafting. 

Shafting  is  supported  in  three  ways,  viz.,  on 
foundation  stones  in  the  floor,  beneath  beams  sus- 
pended from  the  ceiling,  and  to  the  walls  of  the 
mill.  This  necessitates  as  many  different  forms 
of  framework,  known  as  hangers,  plummer-blocks, 
standards,  etc. 

The  simplest  mode  of  supporting  a  range  of 
light  shafting  is  from  the  floor,  and  a  pedestal 
suitable  for  this  purpose  is  shown  in  fig.  139.  It 
consists  of  a  cast  iron  base  plate  and  column,  with 
deep  wings  a  a  cast  on  to  strengthen  it  free  from 
vibration.  The  upper  portion  is  hollowed  out  to 
receive  the  lower  brass  step,  and  the  cap  carrying 


SHAFT   HANGERS   AND   PLUMMER-BLOCKS.    239 

the  upper  step.  When  the  entire  pressure  of  the 
shafting  is  downwards  the  upper  brass  bush  ia 
omitted,  and  the  cap  is  cast  hollow  and  kept  full 
of  grease,  so  as  to  secure  the  most  perfect  lubrica- 
tion of  the  journal  of  the  shaft. 

Fig.  130. 


JB'ig.  140  .shows  a  pedestal  for  bolting  to  a  wall, 
the  chief  difference  being  that  the  cap  is  now  fixed 
on  its  inner  side  by  a  wedge  or  cotter  (c).  In  this 
figure  a  shell  cap  a  is  shown.  If  the  pull  is  up 
wards,  and  two  brasses  be  required,  "  lugs  "  have 
to  be  added  to  the  extremity  of  the  pedestal  and 
cap  for  bolting  the  two  together.  I 

There  are  various  ways  of  suspending  ranges 


240 


MACHINERY  OF  TRANSMISSION. 


of  shafting  from  the  ceiling,  according  to  the 
means  which  exist  for  their  attachment.  If  wooden 
beams,  as  s,  are  present,  the  hanger  has  a  large 


SHAFT  HANGERS  AND   PLUMMER-BLOCKS.     241 
Fig.  141. 


plate  (a),  which  bolts  to  the  side  of  the  beam,  as 
shown  in  figs.  141,  142.  The  caps  are  fixed  by  a 
cotter,  as  in  the  previous  case. 

Figs.  143,  144,  show  a  front  and  side  elevation 
of  another  form  of  hanger  for  attachment  to  wooden 
21 


242 


MACHINERY   OF   TRANSMISSION. 
Pig.  143. 


beams   In  this  case  there  is  provision  for  a  second 
line  of  shafting,  at  right  angles  to,  and  receiving 
motion  from,  the  primary  line.     For  this  purpose 
rig  144,  a  small  plummer-block  is 

bolted  on  to  a  recess  at 
the  side  of  the  hanger. 
The  thrust,  owing  to  the 
pair  of  bevel  wheels  which 
would  be  placed  near  this 
hanger,  is  no  longer  sim- 
ply vertical,  and  hence 
two  brass  steps  are  placed 
for  the  journal  of  the  prin- 
cipal shaft,  with  a  bolt  at 
d,  fig.  1-43,  in  addition  to 
the  cotter,  to  keep  the  cap 
in  its  place. 


SHAFT   HANGERS   AND   PLUHMER-BLOCKS.     243 


Fig.  145  shows  another  form  of  light  hanger 
sometimes  employed  in  weaving  sheds,  and  also 
in  use  for  supporting  shafts  in  fire-proof  mills, 

Fig.  145. 


being  bolted  up  to  the 
under  side  of  the  cast- 
iron  beams,  as  shown  at 
fig.  147. 

Where  greater  strength 
and  firmness  are  re- 
quired, especially  in 
long  hangers  in  which 
there  is  considerable 
leverage,  the  arrange- 
ment shown  in  figs.  146, 
147  is  adopted ;  the 


Pig.  149. 


244 


MACHINERY   OF   TRANSMISSION. 


hanger  in  this  case  is  bolted  to  a  Cast-iron  beam, 
and  by  an  extension  of  the  flange  plate  to  the 
brick  arch,  which  springs  from  the  beam  T,  it 
is  firmly  secured  to  both  beam  and  floor.  At  e 
is  a  screw  for  tightening  the  upper  brass  step  on 
the  shaft. 

Fig.  147. 


==.  Concrete    i=gf==« 


More  complicated  arrangements  are  sometimes 
necessary  where  two  or  three  ranges  of  shafting 
have  to  be  brought  in  connection  with  each  other 


SHAFT   HANGERS   AND   PLUMMER-BLOCKS.    245 

by  means  of  bevel  or  mitre  wheels.  Figs.  148 
and  149,  show  a  front  and  side  elevation  of  this 
arrangement,  which  may  serve  as  a  type  for 
others.  The  hanger  is  attached  to  a  cast-iron 

Fig.  148. 


beam  A,  by  hooked  bolts  with  nuts  beneath  the 
top  plate,  as  shown  at  a  a,  care  being  taken  in  this 
attachment  not  to  weaken  the  flange  of  the  iron  beam 
by  boring  holes  in  it.  Double  brass  steps  are  neces- 
sary in  this  case  for  the  main  line  of  shafting,  and 
also  for  two  smaller  ranges  at  right  angles  to  it, 


246 


MACHINERY   OF   TRANSMISSION. 


which  revolve  in  opposite  directions,  as  shown  at 
fig.  149. 

A  very  frequent  case  in  practice  is  the  connec- 
tion of  two  ranges  of  shafting,  at  right  angles  to 


Fig.  149. 


each  other,  at  the  corner  of  a  room.  This  is 
effected  by  letting  into  the  corner  of  the  building 
a  cast-iron  frame,  commonly  known  as  a  wall-box, 
which  serves  as  a  foundation  for  the  plummer- 
blocks  carrying  the  shafting.  Such  an  arrange- 
ment is  shown  in  fig.  150  in  elevation,  and  in  fig 


SHAFT  HANGERS   AND   PLUMMER-BLOCKS.    247 

151  in  plan.  The  box  w,  w,  w,  is  built  into  the 
wall,  and  bolted  both  to  it  and  to  the  cast-iron 
beam  b.  It  carries  two  pluramer-blocks  on  a 
plate  firmly  supported  by  brackets.  The  wall 

Fig.  150. 


pieces  in  these  two  figures  are  similar,  but  with  a 
slightly  different  arrangement  of  the  plummer- 
blocks. 

Irrespective  of  the  various  forms  of  engaging 
and  disengaging  apparatus,  it  will  be  necessary  to 
consider  the  position,  form,  and  proportions  of  th  i 
wheels  and  shafting  required  in  mills  where  th  * 
power  is  divided  and  widely  distributed.  T.) 
slow  the  enormous  extent  to  which  the  concen- 


248  MACHINERY   OF   TRANSMISSION. 

• 

tration  of  machinery  in  one  building  has  been 
carried,  I  may  mention  that  in  mills  of  my  own 
construction  there  have  been  on  the  average  not 
less  than  450  wheels  and  7,000  feet  of  shafting  in 
motion.  In  the  large  mills  at  Saltaire  there  are 
upwards  of  600  wheels  and  10,000  feet,  or  two 
miles,  of  shafting  distributed  over  an  area  of 

Fig.  151. 


flooring  equivalent  to  12  acres.  In  corn  mills 
and  iron  works,  where  the  machinery  is  more 
closely  connected  with  the  prime  mover,  these 
considerations  are  of  less  importance ;  but  in  fac- 
tories for  the  manufacture  of  textile  fabrics  the 
machinery  covers  a  great  extend  of  surface,  and 


SHAFT   HANGERS   AND   PLUMMER-BLOCKS.    249 

the  greatest  care  is  necessary  in  giving  due  pro- 
portion to  the  transmissive  machinery,  in  order  to 
secure  uniformity  of  motion  at  the  remotest  parts 
of  the  mill. 

In  gearing  a  mill,  the  first  consideration  is  the 
power  of  the  engines,  the  position  of  the  ma- 
chinery to  be  driven,  and  the  strength,  diameter, 
etc.,  of  the  first-motion  shaft,  and  other  requisites 
for  the  transmission  of  motion  in  a  well-geared 
mill.  It  is  upwards  of  twenty  years  since  the  fly- 
wheel was  converted  into  a  first  motion,  and  a 
new  system  of  transmitting  the  power  of  the 
steam  engines  to  the  machinery  of  the  mill  intro- 
duced. Previous  to  that  time  it  was  effected  by 
large  spur-wheels  inside  the  mill,  now  it  is  taken 
direct  from  the  circumference  of  the  fly-wheel.* 
The  advantage  of  this  system  was  the  abolishing 
of  the  cumbrous  first-motion  gearing;  and  the 
requisite  velocity  being  already  present  in  the  fly- 
wheel, it  was  only  necessary  to  cast  it  with  teeth, 
and  to  take  off  the  power  by  a  suitable  pinion  at 
the  level  most  convenient  for  the  purposes  of  the 
mill. 

In  another  place  I  have  given  general  rules  for 
the  pitch,  breadth,  and  strength  of  the  teeth  of 
wheels.  The  Table  (p.  250),  computed  from  ex- 
amples which  have  occurred  in  my  own  practice, 

*  Compare  "Mills  and  Millwork,"  Part  1,  Prime  Movers, 
page  248. 


250 


MACHINERY   OF   TRANSMISSION. 


exhibits  the  best  proportions  of  spur  fly-wheels 
to  secure  durability  of  both  wheel  and  pinion. 

TABLE  9. — DIAMETERS,  PITCH,  VELOCITY,  ETC.,  OF  SPUE 
FLY-WHEELS  OF  THE  NEW  CONSTRUCTION. 


Nominal  power  of 
Steam  Engine. 

Diameter  of 
fly-wheel. 

Pitch  in 

inches. 

Breadth  of 
cog  in 
inches. 

Velocity 
of  pitch  line 
per  minute 
in  feet. 

Iloree-power. 

Ft.     Ins. 

Two  150  =  300 

30         1£ 

4£ 

16 

a 

Single     =    50 

13       3| 

4i 

12 

o 

Two  100  =  200 

24      5 

4 

14 

, 

Two    80  =  160 

23      4 

4 

14 

o 

o 

Two    80  =  160 

22      4 

4 

14 

g    . 

Single            60 

19       OJ 

3| 

12 

tc  _« 

Two    70  =  140 

24      5 

3* 

12 

S  "3 

Two    70  =  140 

22      8^ 

3! 

14 

i| 

Two    50  =  100 

21      0 

12 

•s  a 

Two   40=   80 

21      0 

31 

10 

_o  « 

Two   45=    90 

20      0 

3- 

12 

6>0 

Single            50 

18      2£ 

s| 

12 

.    _g  « 

Two    35=    70 

16       0| 

3i 

10 

c  o 

Single            40 

17     10 

3 

10 

O  lO 

Two    25        50 

13    10 

3 

10 

03  ^ 

Single            25 

8     11£ 

3 

12 

£-0 

a  —  ' 

Two   20=   40 

15      6 

2£ 

7 

Two    25=   50 

15      4£ 

2^ 

8 

OD  m 

Single            25 

15       4£ 

?* 

7 

Hr-T 

Two    18=    36 

13      0 

8 

O 

Single           15 

10      0 

2* 

7 

^ 

Single           18 

17     11 

2 

6 

a 

Single           12 

10      0 

2 

5 

* 

It  will  be  observed  that  the  diameters  of  the 
fly-wheels  are  not  always  proportionate  to  the 
power  of  the  engines,  nor  yet  to  their  respective 
velocities.  In  practice,  it  is  impossible  to  main- 
tain uniformity  in  this  respect,  as,  in  order  to 


PLY  WHEELS  AND   FIRST  MOTIONS.         251 

meet  all  the  requirements  of  manufacture,  it  is 
necessary  to  deviate  from  fixed  principles,  and  to 
approximate  as  near  as  circumstances  will  admit 
to  the  diameters,  weights,  and  velocities  of  wheels, 
as  may  be  found  convenient  to  produce  a  maxi- 
mum effect. 

Of  late  years,  the  speed  of  the  piston  of  factory 
steam  engines  has  been  accelerated  from  240  to 
300,  and  in  some  cases  to  350  feet  per  minute. 
This  united  to  the  increased  pressure  of  steam 
nearly  doubles  the  power  of  the  engines  to  what 
they  were  thirty  years  ago.  The  standard  speed 
of  a  Bolton  and  Watt  7  feet  stroke  engine 
previous  to  that  date,  was  seventeen  and  a  half 
strokes  per  minute. 

In  closing  this  section  of  practical  construction, 
I  may  state  that  the  couplings,  engaging  and  dis- 
engaging gear,  including  the  different  forms  of 
hangers,  fixings,  etc.,  are  taken  from  my  own 
designs,  first  introduced  as  a  substitute  for  the 
cumbrous  attachments  that  were  in  general  use 
previous  to  the  years  1820  and  1823. 

Having  determined  the  diameter,  speed,  and 
strength  of  the  fly-wheel,  the  next  consideration 
is  the  material,  diameter,  etc.,  of  the  main  shaft 
These  are  usually  of  cast-iron,  and  their  diameters 
depending  on  the  power  transmitted  through 
them,  and  the  velocity  at  which  they  revolve,  wil) 
be  found  by  the  tables  and  formulae  already  given 


252  MACHINERY  OF  TRANSMISSION. 

The  distribution  of  the  power  is  usually  effected 
by  a  vertical  shaft,  extending  from  the  bottom 
room,  through  the  various  floors  of  the  mill,  to 
the  top  story  ;  the  power  being  taken  off  at  each 
stage  by  a  pair  of  bevel  wheels.  This  arrange- 
ment, as  shown  in  fig.  152,  represents  one  engine- 
house  with  a  section  of  part  of  one  division  of 
the  mills  at  Saltaire ;  and  this  may  be  considered 
as  a  type  of  other  mills  adapted  for  spinning  and 
similar  purposes. 

It  will  be  observed  that  there  are  four  divisions 
in  the  Saltaire  mills— one  for  the  preparatory 
process,  one  for  the  wool  combing,  another  for 
the  spinning,  and  a  fourth  for  the  weaving.  All 
these  are  driven  by  four  steam  engines,  each  of 
100  nominal  horses'  power,,  but  collectively  dis- 
tributing a  force  through  these  different  depart- 
ments of  upwards  of  1,250  horses. 

On  referring  to  the  drawings,  figs.  152  and 
153,  which  represent  a  cross  and  longitudinal  sec- 
tion of  the  mill,  it  will  be  seen  that  the  vertical 
shaft  AA,  is  driven  direct  from  the  fly-wheel  by 
the  horizontal  shaft  B,  giving  motion  to  the  ma- 
chinery in  each  room  as  it  ascends.  It  is  fixed 
on  a  solid  pier  of  ashlar,  as  shown  at  fig.  154, 
page  257,  and  supported  on  strong  cast-iron 
plates  and  bridgetrees,  firmly  secured  by  bolts  to 
the  foundations  below.  In  each  room  it  is  se- 
curely fixed,  by  cast-iron  frames  and  boxes, 


VERTICAL    SHAFTS. 
Fijr.  152. 


263 


forming  a  recess  for  the  bevel  wheels,  into  the 
wall  which  divides  the  engine-house  and  the 
rooms  above  from  the  mill.  This  wall  is  gene- 
rally made  strong  and  thick,  with  sufficient 
22 


254  MACHINERY   OF  TRANSMISSION. 


JB1 iBI        1111       1U III HI  __  IBL       HI        IHI       1111        'Ill 


VERTICAL   SHAFTS.  255 

weight  to  resist  the  action  of  the  wheels  prepare^. 
to  drive  the  main  lines  of  horizontal  shafts  with  a 
speed  and  force  equivalent  to  the  work  done  in 
each  room.  In  the  case  of  the  Saltaire  mills  this 
is  considerable ;  nearly  300  horses'  power  being 
distributed  through  the  upright  shaft  alone,  the 
remainder  being  carried  off  to  the  loom  shed  by 
a  second  wheel,  working  into  the  bevel  wheel  a, 
on  the  horizontal  shaft  B,  but  not  shown  in  the 
drawing.  It  is  important,  in  mills  where  power- 
ful steam  engines  are  employed,  that  the  founda^ 
tions  and  fixings  to  which  the  main  shafts  are 
attached  are  of  the  most  substantial  description, 
and  the  greatest  precaution  is  necessary  in  order 
to  secure  them  from  vibration,  and  to  render  them 
perfectly  rigid  when  the  whole  force  of  the 
engines  is  applied. 

In  the  Saltaire  mills,  as  in  many  others  for  the 
manufacture  of  cotton,  flax,  and  wool,  the  pre- 
paratory machinery,  such  as  carding,  combing, 
roving,  etc.,  is  generally  driven  by  lines  of  hori- 
zontal shafts,  or  by  a  series  of  cross  shafts, 
branching  off  at  right  angles  from  the  main  line 
extending  down  the  centre  of  the  room,  as  shown 
at  c  c  in  No.  II.  room.  Nos.  III.  and  IY.  rooms 
are  driven  by  the  longitudinal  shaft  in  No.  III. ; 
and  Nos.  V.  and  VI.  by  the  shaft  in  No.  V.  room. 
On  this  plan  it  will  be  noticed  that  the  spinning 
machinery  is  driven  by  iron  pulleys  from  the 
horizontal  shafts,  at  a  velocity  of  nearly  200 


256  MACHINERY   OF   TRANSMISSION. 

Devolutions  per  minute,  and  the  straps  or  belts 
from  those  pulleys  are  directed  by  means  of  guide 
pulleys  to  the  machinery  in  both  rooms.  For 
this  purpose,  iron  boxes  are  inserted  into  the 
arches  supporting  the  floors,  for  the*  admission  of 
the  straps  to  the  machinery  in  the  upper  floor. 

It  will  not  be  necessary  to  give  the  dimensions 
of  the  shafts  in  each  room,  as  these  details  and 
calculations  must  be  left  to  the  judgment  of  the 
millwright,  and  the  nature  of  the  work  they  have 
to  perform.  Suffice  it  to  observe,  that  the  ver- 
tical shaft  A  is  10  inches  diameter  through  the 
first  two  rooms,  8J  inches  through  the  third  room, 
and  6|  inches  to  the  top ;  the  velocity  being  94 
revolutions  per  minute. 

As  respects  the  couplings  for  this  shaft,  we 
may  refer  the  reader  to  the  Table  of  Dimensions 
(page  109)  made  from  couplings  actually  in  use, 
and  which  have  been  found,  by  experiment,  ser- 
viceable in  every  case  where  strength  and  dura- 
bility are  required. 

Great  trouble  is  sometimes  experienced  with 
the  foot  of  the  vertical  shaft,  which,  from  its 
weight  and  the  great  pressure  upon  it,  has  a  ten- 
dency to  heat,  unless  sufficient  bearing-area  is 
allowed,  and  the  parts  kept  thoroughly  lubricated. 
The  general  arrangement  of  the  footsteps  and 
gearing  in  large  mills  is  shown  in  fig.  154 ;  s  s  is 
the  first-motion  shaft,  and  1 1  the  vertical  shaft ; 
a  a  the  bevel  wheel  on  the  former,  and  b  b  the 


VERTICAL  SHAFTS. 


257 


Fig.  154. 


22* 


258  MACHINERY   OF   TRANSMISSON". 

bevel  wheel  on  the  latter;  c  a  plummer-block  for 
the  first  motion-shaft,  and  d  d  the  box  containing 
the  brass  footstep  for  the  vertical  shaft ;  this  box 
rests  on  a  large  base  plate,  bolted  to  the  foun^a- 
tion  stones  and  to  the  wall  of  the  engine-house. 
In  order  to  insure  a  constant  supply  of  oil  to  the 
bearing,  it  is  usual  to  cut  away  nearly  the  whole 
depth  of  the  footstep,  or  that  portion  of  the  brass 
in  the  corner  opposite  to  the  thrust  of  the  bevel 
wheels,  as  shown  in  the  plan,  fig.  154 ;  this  cavity 
is  then  kept  full  of  oil,  and  lubricates  the  shaft 
throughout  at  every  revolution.*  Again,  in  cot- 
ton, woolen,  and  flax  mills,  when  the  first  motion 
and  vertical  shafts  have  been  duly  proportioned 
to  the  work  they  have  to  perform,  it  becomes  ne- 
cessary to  consider  the  diameter,  speeds,  etc.,  of 
the  light  shafting  for  driving  the  machinery  in 
the  different  rooms.  -  The  formula  given  for 
strength,  etc.,  in  a  former  part  of  this  work,  will 
apply  to  this  description  of  gearing  and  mill-work 
where  the  length  does  not  exceed  120  feet.  In 
long  ranges  of  shafts,  of  from  150  to  200  feet  in 
length,  where  the  power  applied  to  the  machinery 
at  the  end  of  the  room  is  considerable,  it  is  essen- 
tially necessary  to  increase  their  strengths  in  order 
to  prevent  torsion  or  twist.  This  is  a  considera- 
tion of  much  importance,  and  requires  careful 

*  The  reader  may  compare  what  is  here  said  of  footsteps 
with  that  in  Mills  and  Millwork,  Part  I.,  pp.  168,  172,  oil 
.the  steps  for  turbine  shafts. 


VERTICAL   SHAFTS.  259 

attention,  as  long  ranges  of  light  shafts  are  very 
elastic — they,  in  some  cases,  effect  nearly  a  com- 
plete revolution  at  the  point  of  imparted  motion 
before  the  extreme  ends  begin  to  move.  The 
result  of  the  power  so  irregularly  transmitted  by 
the  spring  of  the  shafts,  resolves  itself  into  a  series 
of  accelerated  and  retarded  motions  through  the 
whole  line  of  shafts,  and  imparts  to  the  machinery 
in  one-half  of  the  room  a  very  variable  motion. 
Want  of  stiffness  is  a  great  evil  in  long  lines  of 
shafting,  and,  as  we  have  already  observed,  in 
stances  are  not  wanting  in  which  whole  lines  have 
been  removed  entirely  from  this  cause. 

The  transmission  of  power  to  machinery  placed 
at  different  angles  from  the  line  of  shafts,  which 
is  sometimes  the  case  in  old  mills,  has  generally 
been  effected  by  the  universal  joint  A,  fig.  15oi 

Fig.  165. 


which  works  moderately  well  at  an  obtuse  angle, 
but  I  have  always  found  in  my  own  practice  that 
bevel  wheels,  as  at  B,  fig.  156,  are  preferable  and 
more  satisfactory.  They  give  much  less  trouble, 
and  work  with  greater  ease,  than  the  universal 


260  MACHINERY   OF   TRANSMISSION. 

joint.  Other  examples  might  be  given  for  the 
guidance  of  the  practical  millwright ;  but,  having 
to  discuss  these  points  at  greater  -length  when  we 
come  to  treat  of  the  different  kinds  of  mills  and 
different  methods  of  gearing,  we  must  direct  the 

Fig.  156. 


reader  to  those  portions  of  the  work  which  con- 
cern his  own  immediate  practice. 

The  following  table  exhibits  the  diameter  of 
shafts,  length  of  journals,  diameter  and  propor- 
tions of  couplings,  etc.,  derived  from  actual  prac- 
tice, which  may  be  useful  to  the  less  experienced 
millwright  and  engineer: 


PROPORTIONS   OF   SHAFTS. 


261 


TABLE  10. — LENGTH,  DIAMETER,  ETO.,  OF  COUPLINGS,  COUP- 
LINO-BOXES,  ETC. 


Diameter 

Length 

Diameter 

Length 

Length 

Diameter 

Thickness 

of 

of 

of 

of 

of 

of 

of 

Shan. 

Neck. 

Coupling. 

Lap. 

Box. 

Box. 

Metal. 

&lf 

3 

4 

2 

4* 

4* 

1 

l| 

3* 

3 

2i 

5 

5 

1 

2 

4 

N 

2^ 

5* 

6* 

r£ 

21 

*l 

4 

2| 

6 

6 

ii 

2j 

5 

4 

3 

6* 

6f 

if 

2* 

5 

** 

N 

7 

7* 

i* 

3 

5£ 

4J 

3* 

7* 

7* 

i* 

3* 

N 

5 

3| 

8 

8* 

i| 

a* 

«l 

5* 

3J 

N 

8} 

i| 

4 

7 

6 

.4 

8£ 

9* 

if 

4* 

H 

6£ 

4* 

9 

10* 

2 

5 

8 

7l 

5 

10 

111 

,    2 

6* 

8* 

8 

5* 

11 

12| 

2^ 

6 

9 

9 

6 

12 

13* 

2^ 

6£ 

M 

M 

6* 

13 

14f 

2^ 

7 

104 

1W 

•    7 

14 

16 

2| 

7* 

Uj 

111 

9| 

15 

17 

2| 

8 

12 

12 

8 

16} 

18 

3 

8* 

12} 

12^ 

8i 

17 

19 

31 

9 

13£ 

13 

9 

18 

20 

3* 

9* 

14 

13J 

9^ 

18 

21 

3} 

10 

14| 

14 

10 

W* 

22| 

.31 

11 

15 

16 

11 

20 

24 

4 

12 

16 

17* 

12 

21 

26 

4| 

13 

17 

18* 

13 

22 

27* 

** 

INDEX. 


PAOE 

Annular  wheels 64 

Archimedean  screw  creeper 91 

Beam,  the  great,  and  the  crank...    25 
Bevel  wheels  and  bevel  gear 66 

—  skew  bevel  wheels 100 

—  bevel  wheels    preferable  to 
universal  joint 259 

Cafnbs 76 

—  to  find  the  curve  forming  the 
groove  of  a  camb,  so  that  the 
velocity  ratio  of  the  rod   and 
axis  of  the  camb  may  be  con- 
stant     77 

—  to  produce  a  changing  reci- 
procating motion  by  a  combina- 
tion of  the  camb  and  screw 94 

Concentric  wheels 64 

Connectors,  wrapping 40 

Crank  and  great  beam,  the 25 

—  relations  of  crank  and  pis- 
ton..   27 

Crown  wheels 65 

Cutting  machine  of  Messrs.  Peter 
Fairbairn  &  Co.,  of  Leeds Ill 

Detent,  ratchet  wheel  and 89 

Eccentric  wheel,  the 75 

Epicycloidal  teeth  of  wheels 125 

Face  wheel  and  lantern 65 

wheels  and  trundles 108 

Fairbairn,  Messrs.  Pater  &  Co.,  of 
Leeds,  their  cutting  machine....  Ill 

Friction  of  shafting 209 

clutch 226 

cones 227 

— —  coupling 228 

discs 229 

— —  means  adopted  to  lessen  the, 
at  the  foot  of  the  main  vertical 
shaft 255 

Guide  pulleys 47 

Outta  percha,  value  of,  for  straps.  101 

Hangers 238 


Hero  of  Alexandria,  his  mention 
of  toothed  wheels 104 

Idle  wheels 63 

Intermittent  motion  produced  by 
linkwork,  connected  with  a  rat- 
chet-wheel    39 

Involute  teeth  of  wheels 135 

Journals,  length  of. 207 

Lantern,  face-wheel  and 66 

Link-work 22 

— — •  the  crank  and  great  beam....  25 
to  construct  Watt's  parallel 

motion 31 

— —  to  multiply  oscillations  by 

means  of  link-work 34 

— —  to  produce  a  velocity  which 

shall    be   rapidly  retarded   by 

means  of  link-work 36 

—  to  produce  a  reciprocating  in- 
termittent motion  by  means  of 
link-work 37 

—  ratchet-wheel  and  detent 39 

intermittent  motion  produced 

by  link-work  connected  with  a 

ratchet-wheel 39 

Lubrication  of  shafting 213 

Mechanism,  principles  of. 13 

— —  general  views  relative  to  ma- 
chines   13 

definitions  and   preliminary 

expositions 13 

parts  of  a  machine 18 

— —  elementary  forms  of  mechan- 
ism    21 

link-work 22 

the  crank  and  great  beam....    26 

to  construct  Watt's  parallel 

motion 31 

— —  to  multiply  oscillations    by 

means  of  link-work 34 

— —  to  produce  a  velocity  which 
shall  be  rapidly  retarded  by 

means  of  link-work 36 

to  produce  a    reciprocating 

intermittent  motion  by  means 
of  link-work 37 


(263) 


264 


INDEX. 


PAGE 

Mechanism— -continued. 

••       ratchet  wheel  and  detent 89 

—  intermittent  motion  produced 
by  link-work  connected  with  a 
ratchet-wheel... 39 

— —  wrapping  connectors 40 

— —  endless  cord  or  belt 40 

— —  speed  pulleys 44 

guide  pulleys „..  47 

— —  to  prevent  wrapping  connect- 
ors from  slipping 48 

—  systems  of  pulleys 51 

-•*    to  produce  a  varying  Velocity 

ratio  by  means  of  wrapping  con- 
nectors      64 

— —  on  wheel-work  producing  mo- 
tion by  rolling  contact  when 
the  axes  of  motion  are  parallel..  56 

—  idle  wheels 63 

— —  annular  wheels.... 64 

concentric  wheels 64 

•  wheel-work  when   the  axes 
are  not  parallel  to  each  other...    04 

•  face-wheel  and  lantern 65 

crown-wheels 65 

•  to  construct  bevel  wheels  and 
bevel  gear,  when  the  axes  are 

in  the  same  plane 66 

—  to  construct  bevel  gear  when 
the  axes  are  not  in  the  same 
plane .'. 68 

— —  variable  motions  produced  by 
wheel-work  having  rolling  con- 
tact   69 

— —  Roomer's  wheels 71 

— —  intermittent  and  reciprocat- 
ing motions,  produced  by  wheel- 
work  having  rolling  contact 71 

•  •     the  rack  and  pinion 72 

sliding  pieces,  producing  mo- 
tion by  sliding  contact 74 

•  the  wedge,  or  movable   in- 
clined plane 74 

— —  the  eccentric  wheel 75 

•  ciinibs,  wipers,  and  tappets...  76 

—  the  swash  plate 80 

screws,  different  forms  of. 82 

for  cutting  screws 86 

—  to  produce  a  changing  reci- 
procating rectilinear  motion  by 
a  combination  of  the  camb  and 
screw 

^— —  to  produce  a  boring  motion 
by  a  combination  of  the  screw 
and  toothed  wheels 05 

Machinery  of  transmission,  on 99 

Oscillations,  to  multiply  by  means 

of  link-work     34 

Odontograph,  Prof.  Willis's 142 


94 


PA01 

Parallel  motion,  Watt's,   to  con- 
struct      31 

Piston,  relations  of  the  crank  and..    27 

Press,  the  common 86 

Pulleys,  speed 44 

guide 47 

Pulleys,  systems  of. 61 

and  wheels 99 

Pitch  of  wheels 116 

Pinion  from  Kamelli 104 

Plummer-blocks 238 


Rack  and  pinion,  the 72 

Ratchet-wheel  and  detent,  the 39 

intermittent  motion  produced 

by    link-work  connected  with  a 

ratchet-wheel 39 

Reciprocating  intermittent  mo- 
tion, to  produce  a,  by  means  of 

link-work 37 

Rolling  contact,  motion  produced 

by 66 

variable  motions  produced  by 

wheel-work  having  rolling  con- 
tact   , 71 

Ramelli,  pinion- from 104 

Rannie,  Mr.  John,  his  introduc- 
tion of  cast-iron  into  all  the  de- 
tails of  mill-work Ill 

Screws 82 

construction  of  a   belix    or 

screw 82 

——  pitch  of  a  screw 83 

transmission    of  motion    by 

the  screw 83 

solid  screw  and  nut 85 

— — the  common  press 86 

compound  screw 88 

endless  screw 89 

differential  screw 90 

Archimedean  screw  creeper...  91 

— —  mechanism  for  cutting  screws  92 
•'-•    to  produce  a  changing  reci- 
procating rectilinear  motion  by 
a  combination  of  the  camb  and 

screw ..  94 


to  produce  a  boring  motion 

by  a  combination  of  the  screw 

and  toothed  wbeelf 95 

Sliding-pieces,    producing   motion 

by  sliding  contact 74 

the  wedge,  or  movable   In- 
clined plane 74 

the  eccentric  wheel 75 

cambs,  wipers,  and  tappets...  76 

— —  the  swash  plate 80 

screws 82 

—  the  common  press 86 


INDEX. 


265 


PAGE 

Sliding  pieces — continued. 

—  to  produce  a  changing  reci- 
procating motion  by  a  combina- 
tion of  the  camb  and  screw 94 

Speed  pulleys 44 

Swash  plate,  the 80 

Saltaire  mills,  the 252 

Shafts,  on  the  strength  and  pro- 
portion of 175 

1.  The    Material    of    which 

shafting  is  constructed 177 

2.   Transverse  Strain 179 

rules    for    the    strength   of 

shafts 183 

—  resistance  to  flexure;  weights 
producing  a  deflection  of  1-1200 
of  the  length  in  cast-iron  cylin- 
drical shafts 187 

resistance  to  flexure ;  weights 

producing  a.  deflection  of  1-1200 
of  the  length  in  wrought-irou  cy- 
lindrical shafts 188 

deflection  arising  from  the 

weight  of  the  shaft  in  both  cast- 
iron  and  wrought-iron  cylindri- 
cal shafts 189 

3.  Torsion 190 

values  of  modulus  of  torsion 

according  to  Mr.  Bevan 194 

resume  of  experiments  on  cy- 
linders of  circular  section 196 

resume  of  experiments  on  the 

torsion  of  hollow  cylinders  of 
copper 197 

resume  of  experiments  on  the 

torisou  of  elliptical  bars 197 

safe  working  torsion  for  cast- 
iron  shafts 200 

— —  safe  working  torsion  for 
wrought-iron  shafts 201 

diameter  of  wrought-iron 

shafting  necessary  to  transmit 
with  safety  various  amounts  of 
force 205 

4.  Velocity  of  Shafts 204 

5.  Length  of  Journals 207 

6.   Friction 209 

table  of  coefficients  of  friction 

under  pressures  increased  con- 
tinually up  to  limits  of  abra- 
sion   212 

7.  Lubrication 213 

8.  On  Couplings  for  shafts  and 

engaging  and  disengaging  gear..  216 

—  disengaging  and  re-engaging 
gear 221 

Hangers,  Plummer-blocks, 

etc.,  for  carrying  shafting 238 

— —  diameters,  pitch,  velocity, 
etc.,  of  spur  fly-wheels  of  the 
new  construction 250 


PAGE 
Shafts — continued. 

Material,   etc.,   of   the  main 

shafts 251 

vertical  shafts 252 

the  Saltaire  mills 252 

table    of    length,    diameter, 

etc.,  of  couplings,  coupling-box- 
es, etc 261 

Skew  bevel-wheels 160 

Smeatoii's  introduction  of  cast-iron 

gearing  in  place  of  wood 107 

Spur  gearing 114 

Straps     compared     with     geared 

wheel-work 100 

— —  materials  of  which  straps  are 

made 101 

strength  of  straps 102 

table  of   the   least  width  of 

straps  for  transmitting  various 
amounts  of  work  over  different 
pulleys 103 

Tappets,  or  wipers 76 

Tables  relating  to  straps.  See 
Wrapping  connectors : 

wheels.    See  Wheel-work. 

shafts.    See  Shafts. 

Teeth  of  wheels.  See  Wheel-work. 
Toothed  wheels,  history  of. 103 

Universal  joint,  bevel  wheels  pre- 
ferable to 259 

Velocity,  to  produce  a,  which  shall 
be  rapidly  retarded,  by  means 
of  link-work 36 

Watt's  parallel  motion,  to  con- 
struct   31 

Wedge,  the,  or  movable  inclined 
plane 74 

Wheel-work,  producing  motion  of, 
by  rolling  contact  when  the 
axes  of  motion  are  parallel 56 

idle-wheels 63 

annular  wheels 64 

concentric  wheels 64 

—  when  the  axes  are  not  paral- 
lel to  each  other 64 

— —  face- wheel  and  lantern 65 

crown  wheels 65 

to  construct  bevel  wheels,  or 

bevel  gear,  when  the  axes  are 
in  the  same  plane 6(j 

to  construct  bevel  gear  when 

the  axes  are  not  in  the  same 
plane 68 

variable  motions  produced  by 

wheel-work  having  rolling  con- 
tact    69 

Roemer's  wheels 71 


266 


IXDEX. 


PAGE 

Wheel-work — continued. 
——  intermittent  and  reciprocat- 
ing motions,  produced  by  wheel- 
work  having  rolling  contact 71 

to  produce  a  boring  motion 

by  a  combination  of  the  screw 

and  toothed  wheels 95 

Wipers,  or  tappets 76 

Wrapping  connectors 40 

endless  cord  or  belt 40 

speod  pulleys 44 

guide  pulloys 47 

to  prevent  wrapping  connect- 
ors from  slipping 48 

systems  of  pulleys 51 

to  produce  a  varying  velocity 

ratio    by    means   of    wrapping 

connectors 54 

Wheel-work,  power  of  straps  com- 
pared with  that  of  geared 

wheel-work 100 

history  of  toothed  wheels 103 

Hero  of  Alexandria 104 

Ramelli 104 

introduction  of  cast-iron  gear- 
ing   107 

face-wheels  and  trundles 10S 

bevel  wheels 66 

causes  of  the  defects  of  wheel- 
work 109 

cutting  machine  of  Messrs. 

Peter'Fairbairn  &  Co.,  of  Leeds..  Ill 

definitions  of  spur  gearing....  114 

the  pitch  of  wheels 116 

table  of  the  relation  of  diame- 
ter, pitch,  and  number  of  teeth, 
for  wheels  of  from  }/£  inch  to  5 
inches  pitch,  and  from  12  to  200 

teeth 123 

the  principles  which  deter- 
mine the  proper  form  of  the 

teeth  of  wheels 124 

—  construction  of  epicycloidal 
toeth 129 


PAO« 
Wheel-work — continued. 

epicycloidal  teeth 125 

the  rack 135 

involute  teeth 135 

Prof.  Willis's  method  of  strik- 
ing the  teeth  of  wheels 140 

Prof.  Willis,  his  odontograph.  142 

general  form  and  proportion 

of  toothed  wheels 146 

table  giving  the  proportions 

of  the  teeth  of  wheels  in  inches 

and  thirty  seconds  of  an  inch 156 

table  of  proportions  of  teeth 

of  wheels  for  average  practice...  154 

skew  bevel  wheels 160 

worm  and  wheel 163 

strength    of    the    teeth    of 

wheels 165 


table  of  thickness,   breadth 

and  pitch  of  teeth  of  wheels 168 

table  of  the  relation  of  horses' 

power  transmitted,  and  velocity 
at  the  pitch  circle,  to  pressure 

on  teeth .'. 172 

— —  table  showing  the  pitch  and 
thickness  of  teeth  to  transmit  a 
given  number  of  horses'  power 

at  different  velocities 173 

table  showing  the  breadth  of 

teeth  required  to  transmit  differ- 
ent amounts  of  force  at  a  uni- 
form pressure  of  400  Ibs  per  inch.  174 

Wheels  and  pulleys 99 

Willis,  Prof.,  his  method  of  striking 

the  teeth  of  wheels 140 

his  odoutograjih 142 

Worm  and  wheel,  the 163 

Wrapping  connectors 99 

power  of    straps    compared 

with  that  of  geared  wheel-work.  100 

table    of    the    approximate 

width  of  leather  straps  in  inches 
necessary  to  transmit  any  num- 
ber of  horses'  power 103 


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chanic. '  With  over  One  Hundred  Illustrations.  12mo.  $2  50 


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•pOOTH.— MARBLE  WORKER'S  MANUAL  : 

Containing  Practical  Information  respecting  Marbles  in  gene- 
ral, their  Cutting,  Working,  and  Polishing ;  Veneering  of 
Marble  ;  Mosaics ;  Composition  and  Use  of  Artificial  Marble, 
Stuccos,  Cements,  Receipts,  Secrets,  etc.  etc.  Translated 
from  the  French  by  M.  L.  BOOTH.  With  an  Appendix  con- 
cerning American  Marbles.  12mo.,  cloth  .  $1  50 

•DOOTH  AND  MORFIT.—  THE  ENCYCLOPEDIA  OF  CHEMISTRY, 
™    PRACTICAL  AND  THEORETICAL  : 

Embracing  its  application  to  the  Arts,  Metallurgy,  Mineralogy, 
Geology,  Medicine,  and  Pharmacy.  By  JAMES  C.  BOOTH, 
Melter  and  Refiner  in  the  United  States  Mint,  Professor  of 
Applied  Chemistry  in  the  Franklin  Institute,  etc.,  assisted  by 
CAMPBELL  MORFIT,  author  of  "Chemical  Manipulations,"  etc. 
Seventh  edition.  Complete  in  one  volume,  royal  8vo.,  978 
pages,  with  numerous  wood-cuts  and  other  illustrations.  S|55  00 

•pOWDITCH.— ANALYSIS,  TECHNICAL  VALUATION,  PURLFI- 

D    CATION,  AND  USE  OF  COAL  GAS : 

By  Rev.  W.  R.  BOWDITCH.  Illustrated  with  wood  engrav- 
ings. 8vo.  .  . $6  50 

•pOX.— PRACTICAL  HYDRAULICS : 

A  Series  of  Rules  and  Tables  for  the  use  of  Engineers,  etc. 
By  THOMAS  Box.  12mo. $2  00 

•pUCKMASTER.— THE  ELEMENTS  OF  MECHANICAL  PHYSICS  : 
By  J.  C.  BUCKMASTER,  late  Student  in  the  Government  School 
of  Mines ;  Certified  Teacher  of  Science  by  the  Department  of 
Science  and  Art ;  Examiner  in  Chemistry  and  Physics  in  the 
Royal  College  of  Preceptors ;  and  late  Lecturer  in  Chemistry 
and  Physics  of  the  Royal  Polytechnic  Institute.  Illustrated 
with  numerous  engravings.  In  one  vol.  12mo.  .  $2  00 

DULLOCK— THE  AMERICAN  COTTAGE  BUILDER : 

A  Series  of  Designs,  Plans,  and  Specifications,  from  $200  to 
to  §20,000  for  Homes  for  the  People ;  together  with  Warm- 
ing, Ventilation,  Drainage,  Painting,  and  Landscape  Garden- 
ing. By  JOHN  BULLOCK,  Architect,  Civil  Engineer,  Mechani- 
cian, and  Editor  of  "  The  Rudiments  of  Architecture  and 
Building,"  etc.  Illustrated  by  75  engravings.  In  one  vol. 
8vo $3  50 


HENRY  CAREY  BAIRB'S    CAT.VLOGTTE. 


TTJLLOCK.  —  THE    RUDIMENTS     OF     ARCHITECTURE    AND 
S    BUILDING : 

For  the  use  of  Architects,  Builders,  Draughtsmen,  Machin- 
ists, Engineers,  and  Mechanics.  Edited  by  JOHN  BULLOCK, 
author  of  "The  American  Cottage  Builder."  Illustrated  by 
250  engravings.  In  one  volume  8vo.  .  .  $3  50 

•pURGH.— PEACTICAL  ILLUSTRATIONS   OF  LAND   AND  MA- 

a    BINE  ENGINES : 

Showing  in  detail  the  Modern  Improvements  of  High  and  Low 
Pressure,  Surface  Condensation,  and  Super-heating,  together 
•with  Land  and  Marine  Boilers.  By  N.  P.  BUKGII,  Engineer. 
Illustrated  by  twenty  plates,  double  elephant  folio,  with  text. 

$21  00 

•nURGH.— PRACTICAL    RULES    FOR  TH3   PROPORTIONS    OF 

D    MODERN  ENGINES  AND   BOILERS  FOR  LAND  AND  MA- 
RINE PURPOSES. 
By  N.  P.  BURG u,  Engineer.     12mo.  .         .         .     $2  00 

•pURGH.— THE  SLIDE-VALVE  PRACTICALLY  CONSIDERED : 
By  N.  P.  BURGH,  author  of  "  A  Treatise  on  Sugar  Machinery," 
"Practical  Illustrations  of  Land  and  Marine  Engines,"  "A 
Pocket-Book  of  Practical  Rules  for  Designing  Land  and  Ma- 
rine Engines,  Boilers,"  etc.  etc.  etc.  Completely  illustrated. 
12mo §2  00 

T)YRN.— THE  COMPLETE  PRACTICAL  BREWER  : 

Or,  Plain,  Accurate,  and  Thorough  Instructions  in  the  Art  of 
Brewing  Beer,  Ale,  Porter,  including  the  Process  of  making 
Bavarian  Beer,  all  the  Small  Beers,  such  as  Root-beer,  Ginger- 
pop,  Sarsaparilla-beer,  Mead,  Spruce  beer,  etc.  etc.  Adapted 
to  the  use  of  Public  Brewers  and  Private  Families.  By  M.  LA 
FAYETTE  BTRN,  M.  D.  With  illustrations.  12mo.  $1  25 

TDYRN.— THE  COMPLETE  PRACTICAL  DISTILLER  : 

Comprising  the  most  perfect  and  exact  Theoretical  and  Prac- 
tical Description  of  the  Art  of  Distillation  and  Rectification ; 
including  all  of  the  most  recent  improvements  in  distilling 
apparatus;  instructions  for  preparing  spirits  from  the  nume- 
rous vegetables,  fruits,  etc. ;  directions  for  the  distillation  and 
preparation  of  all  kinds  of  brandies  and  other  spirits,  spiritu- 
ous and  other  compounds,  etc.  etc. ;  all  of  which  is  so  simpli- 
fied that  it  is  adapted  not  only  to  the  use  of  extensive  distil- 
lers, but  for  every  farmer,  or  others  who  may  wish  to  engage 
in  the  art  of  distilling.  By  M.  LA  FAYETTE  BVRN,  M.  D. 
With  numerous  engravings.  la  one  volume,  12mo.  $1  50 


HENRY  CAREY  BAIRD'S  CATALOGUE.  f- 

BYRNE.— POCKET  BOOK  FOE  RAILROAD   AND   CIVIL  ENGi- 
™     NEESS : 

Containing  New,  Exact,  and  Concise  Methods  for  Laying  out 
Railroad  Curves,  Switches,  Frog  Angles  and  Crossings;  the 
Staking  out  of  work;  Levelling;  the  Calculation  of  Cut- 
tings; Embankments;  Earth-work,  etc.  By  OLIVER  BYRXE. 
Illustrated,  18mo .  $1  25 

•DYRNE.— THE  HANDBOOK  FOR  THE  ARTISAN,  MECHANIC, 
•      AND  ENGINEER : 

By  OLIVER  BYRNE.  Illustrated  by  11  large  plates  and  185 
Wood  Engravings.  8vo $5  00 

TDYRNE.— THE  ESSENTIAL  ELEMENTS   OF  PRACTICAL    ME- 
a    CHANICS : 

For  Engineering  Students,  based  on  the  Principle  of  Work. 
By  OLIVER  BYRNE.  Illustrated  by  Numerous  Wood  Engrav- 
ings, 12mo $3  63 

•pYRNE.— THE  PRACTICAL  METAL-WORKER'S  ASSISTANT: 
Comprising  Metallurgic  Chemistry ;  the  Arts  of  Working  all 
Metals  and  Alloys;  Forging  of  Iron  and  Steel;  Hardening  and 
Tempering;  Melting  and  Mixing;  Casting  and  Founding; 
Works  in  Sheet  Metal ;  the  Processes  Dependent  on  the 
Ductility  of  the  Metals ;  Soldering ;  and  the  most  Improved 
Processes  and  Tools  employed  by  Metal-Workers.  With  the 
Application  of  the  Art  of  Electro-Metallurgy  to  Manufactu- 
ring Processes ;  collected  from  Original  Sources,  and  from  the 
Works  of  Holtzapffel,  Bergeron,  Leupold,  Plumier,  Napier,  and 
others.  By  OLIVER  BYRNE.  A  New,  Revised,  and  improved 
Edition,  with  Additions  by  John  Scoffern,  M.  B  ,  William  Clay, 
Wm.  Fairbairn,  F.  R.  S.,  and  James  Napier.  With  Five  Hun- 
dred and  Ninety-two  Engravings ;  Illustrating  every  Branch 
•  of  the  Subject.  In  one  volume,  8vo.  652  pages  .  $7  00 

TDYRNE.— THE  PRACTICAL  CALCULATOR  : 

For  the  Engineer,  Mechanic,  Manufacturer  of  Engine  Work, 
Naval  Architect,  Miner,  and  Millwright.  By  OLIVER  BYRNE. 
1  volume,  8vo.,  nearly  600  pages  .  .  .  .  $4  50 

plBINET  MAKER'S  ALBTJM  OF  FURNITURE: 

Comprising  a  Collection  of  Designs  for  the  Newest  and  Mist 
Elegant  Styles  of  Furniture.  Illustrated  by  Forty  eight  Large 
and  Beautifully  Engraved  Plates.  In  one  volume,  oblong 

$5  00 


HENRY  CAREY  BAIRD'S  CATALOGUE. 


PIALVERT.— LECTURES  ON   COAL-TAR  COLORS,  AND  ON  RE- 
U     CENT  IMPROVEMENTS  AND  PROGRESS  IN  DYEING  AND 
CALICO  PRINTING: 

Embodying  Copious  Notes  taken  at  the  last  London  Interna- 
tional Exhibition,  and  Illustrated  with  Numerous  Patterns  of 
Aniline  and  other  Colors.  By  F.  GRACE  CALVERT,  F.  R.  S., 
F.  C.  S.,  Professor  of  Chemistry  at  the  Royal  Institution,  Man- 
chester, Corresponding  Member  of  the  Royal  Academies  of 
Turin  and  Rouen ;  of  the  Pharmaceutical  Society  of  Paris ; 
Socie"t£  Industrielle  de  Mulhouse,  etc.  In  one  volume,  8vo., 
cloth  .  . $1  50 

nAMPIN.— A  PRACTICAL  TREATISE  ON  MECHANICAL   EN- 
U     GINEERING: 

Comprising  Metallurgy,  Moulding,  Casting,  Forging,  Tools, 
Workshop  Machinery,  Mechanical  Manipulation,  Manufacture 
of  Steam-engines,  etc.  etc.  With  an  Appendix  on  the  Ana- 
lysis of  Iron  and  Iron  Ores.  By  FRANCIS  CAMPIN,  C.  E.  To 
which  are  added,  Observations  on  the  Construction  of  Steam 
Boilers,  and  Remarks  upon  Furnaces  used  for  Smoke  Preven- 
tion ;  with  a  Chapter  on  Explosions.  By  R.  Armstrong,  C.  E., 
and  John  Bourne.  Rules  for  Calculating  the  Change  Wheels 
for  Screws  on  a  Turning  Lathe,  and  for  a  Wheel-cutting 
Machine.  By  J.  LA  NICCA.  Management  of  Steel,  including 
Forging,  Hardening,  Tempering,  Annealing,  Shrinking,  and 
Expansion.  And  the  Case-hardening  of  Iron.  By  G.  EDE. 
8vo.  Illustrated  with  29  plates  and  100  wood  engravings. 

$G  00 

pAMPlN.— THE    PRACTICE    OF  HAND-TURNING  IN  WOOD, 
U     IVORY,  SHELL,  ETC. : 

With  Instructions  for  Turning  such  works  in  Metal  as  may  be 
required  in  the  Practice  of  Turning  Wood,  Ivory,  etc.  Also, 
an  Appendix  on  Ornamental  Turning.  By  FRANCIS  CAMPIN; 
with  Numerous  Illustrations,  12mo.,  cloth  .  .  §3  00 

p  APRON  DE  DOLE.— DUSSAUCE.— SLUES  AND  CARMINES  OF 
^     INDIGO. 

A  Practical  Treatise  on  the  Fabrication  of  every  Commercial 
Product  derived  from  Indigo.  By  FELICIEN  CAPRON  DE  DOLE, 
Translated,  with  important  additions,  by  Professor  H.  Dus- 
SAECE.  12mo.  $2  50 


HENRY  CAREY  BAIRD'S  CATALOGUE. 


pft.REY.— THE  WORKS  OF  HENRY  C.  CAREY: 

CONTRACTION  OR  EXPANSION?  REPUDIATION  OR  RE- 
SUMPTION? Letters  to  Hon.  Hugh  McCulloch.  8vo.  38 

FINANCIAL  CRISES,  their  Causes  and  Effects.     8vo.  paper 

25 

HARMONY   OF   INTERESTS;    Agricultural,    Manufacturing, 

and  Commercial.     8vo.,  paper §1  00 

Do.  do.  cloth          .         .         .     $1  50 

LETTERS  TO  THE  PRESIDENT  OF  THE  UNITED  STATES. 
Paper  .........  75 

MANUAL  OF  SOCIAL  SCIENCE.  Condensed  from  Carey's 
"Principles  of  Social  Science."  By  KATE  McKEAN.  1  vol. 
12mo $2  25 

MISCELLANEOUS  WORKS:  comprising  "Harmony  of  Inter- 
ests," "Money,"  "Letters  to  the  President,"  "French  and 
American  Tariffs,"  "Financial  Crises,"  "The  Way  to  Outdo 
England -without  Fighting  Her,"  "Resources  of  the  Union," 
"The  Public  Debt,"  "Contraction  or  Expansion,"  "Review 
of  the  Decade  1857 — 'G7,"  "Reconstruction,"  etc.  etc.  1  vol. 
8vo.,  cloth $4  50 

MONEY:  A  LECTURE  before  the  N.  Y.  Geographical  and  Sta- 
tistical Society.  8vo.,  paper  .....  25 

PAST,  PRESENT,  AND  FUTURE.     8vo.  .         .         .     $2  50 

PRINCIPLES  OF  SOCIAL  SCIENCE.     3  volumes  8vo.,  cloth 

$10  00 

REVIEW  OF  THE  DECADE  1857— '67.     8vo.,  paper  38 

RECONSTRUCTION:  INDUSTRIAL,  FINANCIAL,  AND  PO- 
LITICAL. Letters  to  the  Hon.  Henry  Wilson,  U.  S.  S.  Svo. 
paper .  38 

THE  PUBLIC  DEBT,  LOCAL  AND  NATIONAL.  How  to 
provide  for  its  discharge  while  lessening  the  burden  of  Taxa- 
tion. Letter  to  David  A.  Wells,  Esq.,  U.  S.  Revenue  Commis- 
sion. 8vo.,  paper  .......  25 

THE  RESOURCES  OF  THE  UNION.  A  Lecture  read,  Dec. 
1865,  before  the  American  Geographical  and  Statistical  So- 
ciety, N.  Y.,  and  before  the  American  Association  for  the  Ad- 
vancement of  Social  Science,  Boston  ...  25 

THE  SLAVE  TRADE,  DOMESTIC  AND  FOREIGN;  Why  it 
Exists,  and  How  it  may  be  Extinguished.  12mo.,  cloth  $150 


8  HENRY  CAREY  BAIRD'S  CATALOGUE. 

THE  WAY  TO  OUTDO  ENGLAND  WITHOUT  FIGHTING 
HER.  Letters  to  the  Hon.  Schuyler  Colfax,  Speaker  of  the 
House  of  Representatives  United  States,  on  "The  Paper  Ques- 
tion," "The  Farmer's  Question,"  "The  Iron  Question,"  "The 
Railroad  Question,"  and  "The  Currency  Question."  8vo., 
paper 75 

PHEVALIER.-THE  PHOTOGRAPHIC  STUDENT. 

A  Complete  Treatise  on  the  Theory  and  Practice  of  Photo- 
graphy. Translated  from  the  French  of  A.  CHEVALIER.  Il- 
lustrated by  numerous  engravings.  (In  press.) 

pLOUGH.— THE  CONTRACTOR'S    MANUAL    AND    BUILDER'S 
V     PRICE-BOOK : 

Designed  to  elucidate  the  method  of  ascertaining,  correctly, 
the  value  and  Quantity  of  every  description  of  Work  and  Ma- 
terials used  in  the  Art  of  Building,  from  their  Prime  Cost  in 
any  part  of  the  United  States,  collected  from  extensive  expe- 
rience and  observation  in  Building  and  Designing;  to  which 
are  added  a  large  variety  of  Tables,  Memoranda,  etc.,  indis- 
pensable to  all  engaged  or  concerned  in  erecting  buildings  of 
any  kind.  By  A.  B.  CLOCGH,  Architect,  24mo.,  cloth  75 

pDLBURN.— THE  GAS-WORKS  OF  LONDON: 

Comprising  a  sketch  of  the  Gas-works  of  the  city,  Process  of 
Manufacture,  Quantity  Produced,  Cost,  Profit,  etc.  By  ZERAH 
COLBUBN.  8vo.,  cloth 75 

rOLBURN.— THE  LOCOMOTIVE  ENGINE: 

Including  a  Description  of  its  Structure,  Rules  for  Estimat- 
ing its  Capabilities,  and  Practical  Observations  on  its  Construc- 
tion and  Management.  By  ZERAH  COLBURX.  Illustrated.  A 
new  edition.  12mo. $1  25 

pOLBURN  AND  MAW.— THE  WATER- WORKS  OF  LONDON : 
Together  with  a  Series  of  Articles  on  various  other  Water- 
works.    By  ZERAH  COLBURN  and  W.  MAW.     Reprinted  from 
"Engineering."     In  one  volume,  8vo.        .  $t  00 

TJ4.GTJERREOTYPIST  AND  PHOTOGRAPHER'S  COMPANION: 
•^     12mo.,  cloth $1  25 

•nflLVIS.— A  TREATISE  ON    HARNESS,   SADDLES,   AND   BRI- 
•^     DLES  : 

Their  History  and  Manufacture  from  the  Earliest  .Times  down 
to  the  Present  Period.  By  A.  DAVIS,  Practical  Saddler  and 
Harness  Maker.  (In  press.) 


HENRY  CAREY  BAIRD'S  CATALOGUE. 


TlESSOYE.— STEEL,  ITS  MANUFACTTJKE,    PROPERTIES,  AND 

•*•'     USE. 

By  J.  B.  J.  DESSOTE,  Manufacturer  of  Steel ;  with  an  Intro- 
duction and  Notes  by  ED.  GRATEN,  Engineer  of  Mines. 
Translated  from  the  French.  In  one  volume,  12mo.  (In  press.) 

•niRCKS  —  PERPETUAL  MOTION : 

Or  Search  for  Self-Motive  Power  during  the  17th,  18th,  and 
19th  centuries.  Illustrated  from  various  authentic  sources  in 
Papers,  Essays,  Letters,  Paragraphs,  and  numerous  Patent 
Specifications,  with  an  Introductory  Essay  by  HENRY  DIRCKS, 
C.  E.  Illustrated  by  numerous  engravings  of  machines. 
12mo.,  cloth  .  .  .  .  .  .  .  .  $3  50 

TlIXON.— THE  PEACTICAL  MILLWRIGHT'S  AND  ENGINEER'S 

**     GUIDE : 

Or  Tables  for  Finding  the  Diameter  and  Power  of  Cogwheels  ; 
Diameter,  Weight,  and  Power  of  Shafts ;  Diameter  and  Strength 
of  Bolts,  etc.  etc.  By  THOMAS  DIXON.  12mo.,  cloth.  $1  50 

T\UNC  AN.— PRACTICAL  SURVEYOR'S  GUIDE: 

Containing  the  necessary  information  to  make  any  person,  of 
common  capacity,  a  finished  land  surveyor  without  the  aid  of 
a  teacher.  By  ANDREW  DUNCAN.  Illustrated.  12mo.,  cloth. 

$1  25 

TJUSSAUCE— A  NEW  AND    COMPLETE    TREATISE    ON  THE 
**     ARTS  OF  TANNING,  CURRYING,  AND  LEATHER  DRESS- 
ING: 

Comprising  all  the  Discoveries  and  Improvements  made  in 
France,  Great  Britain,  and  the  United  States.  Edited  from 
Notes  and  Documents  of  Messrs.  Sallerou,  Grouvelle,  Duval, 
Dessables,  Labarraque,  Payen,  Rene",  De  Fontenelle,  Mala- 
peyre,  etc.  etc.  By  Prof.  H.  DUSSAUCE,  Chemist.  Illustrated 
by  212  wood  engravings.  8vo $10  00 

T)USSAUCE.— A  GENERAL  TREATISE  ON  THE  MANUFACTURE 
"     OF  EVERY  DESCRIPTION  OF  SOAP : 

Comprising  the  Chemistry  of  the  Art,  with  Remarks  on  Alka- 
lies, Saponifiable  Fatty  Bodies,  the  apparatus  necessary  in  a 
Soap  Factory,  Practical  Instructions  on  the  manufacture  of 
the  various  kinds  of  Soap,  the  assay  of  Soaps,  etc.  etc.  Edited 
from  notes  of.  Larm£,  Fontenelle,  Malapeyre,  Dufour,  and 
others,  with  large  and  important  additions  by  Professor  H. 
DUSSAUCE,  Chemist.  Illustrated.  In  one  volume,  8vo.  (In 
press.) 


10  HENRY  CAREY  BAIRD'S  CATALOGUE. 

•nUSSAUCE.— • A  PRACTICAL  GUIDE  FOE  THE  PERFUMER: 
Being  a  New  Treatise  on  Perfumery  the  most  favorable  to  the 
Beauty  without  being  injurious  to  the  Health,  comprising  a 
Description  of  the  substances  used  in  Perfumery,  the  Form- 
ulae of  more  than  one  thousand  Preparations,  such  as  Cosme- 
tics, Perfumed  Oils,  Tooth  Powders,  Waters,  Extracts,  Tinc- 
tures, Infusions,  Vinaigres,  Essential  Oils,  Pastels,  Creams, 
Soaps,  and  many  new  Hygienic  Products  not  hitherto  described. 
Edited  from  Notes  and  Documents  of  Messrs.  Debay,  Lunel, 
etc.  With  additions  by  Professor  II.  DUSSAUCE,  Chemist.  (In 
press,  shortly  to  be  issued.) 

TJUSSAUCE.— PRACTICAL  TREATISE  ON  THE  FABRICATION 

•^     OF  MATCHES,   GUN  COTTON,  AND  FULMINATING  POW- 
DERS. 
By  Professor  II.  DUSSAUCE.     12mo.  .         .         .     $3  00 

TjUSSAUCE.— TREATISE    ON    THE   COLORING  MATTERS  DE- 

r*     EIVED  FROM  COAL  TAR  : 

Their  Practical  Application  in  Dyeing  Cotton,  Wool,  and  Silk; 
the  Principles  of  the  Art  of  Dyeing  and  of  the  Distillation  of 
Coal  Tar,  with  a  Description  of  the  most  Important  New  Dyes 
now  in  use.  By  Prof.  H.  DUSSAUCE.  12mo.  .  $3  00 

T\YER  AND  COLOR-MAKER'S  COMPANION  : 

Containing  upwards  of  two  hundred  Receipts  for  making  Co- 
lors, on  the  most  approved  principles,  for  all  the  various  styles 
and  fabrics  now  in  existence ;  with  the  Scouring  Process,  and 
plain  Directions  for  Preparing,  Washing-off,  and  Finishing  the 
Goods.  In  one  vol.  12mo.  .  .  .  .  .  $1  25 

•DASTON.— A  PRACTICAL  TREATISE  ON  STREET   OR  HORSE- 

**     POWER  RAILWAYS : 

Their  Location,  Construction,  and  Management ;  with  General 
Plans  and  Rules  for  their  Organization  and  Operation ;  toge- 
ther with  Examinations  as  to  their  Comparative  Advantages 
over  the  Omnibus  System,  and  Inquiries  as  to  their  Value  for 
Investment ;  including  Copies  of  Municipal  Ordinances  relat- 
ing thereto.  By  ALEXANDER  EASTON,  C.  E.  Illustrated  by  23 
plates,  8vo.,  cloth $2  00 

•PRNI— COAL  OIL  AND  PETROLEUM : 

Their  Origin,  History,  Geology,  and  Chemistry ;  with  a  view  of 
their  importance  in  their  bearing  on  National  Industry.  By 
Dr.  HENRI  EKNI,  Chief  Chemist,  Department  of  Agriculture. 
]2mo $2  50 


HENRY  CAREY  BAIRD'S  CATALOGUE.  11 


— THE  THEORETICAL  AND  PRACTICAL  CHEMISTRY  OF 

*  FERMENTATION : 

Comprising  the  Chemistry  of  Wine,  Beer,  Distilling  of  Liquors; 
with  the  Practical  Methods  of  their  Chemical  Examination, 
Preservation,  and  Improvement — such  as  Gallizing  of  Wines. 
With  an  Appendix,  containing  well-tested  Practical  Rules  and 
Receipts  for  the  manufacture,  etc.,  of  all  kinds  of  Alcoholic 
Liquors.  By  HENRY  ERNI,  Chief  Chemist,  Department  of 
Agriculture.  (In  press.) 

•PAIRBAIRN.— THE  PRINCIPLES  OF  MECHANISM  AND  MA- 

*  CHINERY  OF  TRANSMISSION : 

Comprising  the  Principles  of  Mechanism,  Wheels,  and  Pulleys, 
Strength  and  Proportions  of  Shafts,  Couplings  of  Shafts,  and 
Engaging  and  Disengaging  Gear.  By  WILLIAM  FAIRBAIRN, 
Esq.,  C.  E.,  LL.  D.,  F.  R.  S.,  F.  G.  S.,  Corresponding  Member 
of  the  National  Institute  of  France,  and  of  the  Royal  Academy 
of  Turin  ;  Chevalier  of  the  Legion  of  Honor,  etc.  etc.  Beau- 
tifully illustrated  by  over  150  wood-cuts.  In  one  volume  12mo. 

$2  50 

pAIRBAIRN.— PRIME-MOVERS : 

Comprising  the  Accumulation  of  Water-power ;  the  Construc- 
tion of  Water-wheels  and  Turbines ;  the  Properties  of  Steam ; 
the  Varieties  of  Steam-engines  and  Boilers  and  Wind-mills. 
By  WILLIAM  FAIRBAIRN,  C.  E.,  LL.  D.,  F.  R.  S.,  F.  G.  S.  Au- 
thor of  "Principles  of  Mechanism  and  the  Machinery  of  Trans- 
mission." With  Numerous  Illustrations.  In  one  volume.  (la 
press.) 

pLAMM— A  PRACTICAL  GUIDE  TO  THE  CONSTRUCTION  OF 

*  ECONOMICAL  HEATING  APPLICATIONS  FOR  SOLID  AND 

GASEOUS  FUELS : 

With  the  Application  of  Concentrated  Heat,  and  on  Waste 
Heat,  for  the  Use  of  Engineers,  Architects,  Stove  and  Furnace 
Makers,  Manufacturers  of  Fire  Brick,  Zinc,  Porcelain,  Glass, 
Earthenware,  Steel,  Chemical  Products,  Sugar  Refiners,  Me- 
tallurgists, and  all  others  employing  Heat.  By  M.  PIERRE 
FLAMM,  Manufacturer.  Illustrated.  Translated  from  the 
French.  One  volume,  12mo.  (In  press.) 

niLBART.— A  PRACTICAL  TREATISE  ON  BANKING: 

By  JAMES  WILLIAM  GILBART.  To  which  is  added:  THE  NA- 
TIONAL BANK  ACT  AS  NOW  (18C8)  IN  FORCE.  8vo.  $4  50 


12  HENRY  CAREY  BAIRD'S  CATALOGUE. 


H  OTHIC  ALBUM  FOR  CABINET  MAKERS : 

Comprising  a  Collection  of  Designs  for  Gothic  Furniture.  Il- 
lustrated by  twenty-three  large  and  beautifully  engraved 
plates.  Oblong §3  00 

QRANT.— BEET-ROOT    SUGAR    AND   CULTIVATION   OF  THE 
BEET : 
By  E.  B.  GRANT.     12mo.  .         .         .         .     .    .    fl  25 

rTREGORY.— MATHEMATICS  FOR  PRACTICAL  MEN  : 

Adapted  to  the  Pursuits  of  Surveyors,  Architects,  Mechanics, 
and  Civil  Engineers.  By  OLINTHUS  GREGORY.  8vo.,  plates, 
cloth $3  00 

nRISWOLD.— RAILROAD  ENGINEER'S  POCKET  COMPANION. 

Comprising  Rules  for  Calculating  Deflection  Distances  and 
Angles,  Tangential  Distances  and  Angles,  and  all  Necessary 
Tables  for  Engineers;  also  the  art  of  Levelling  f:om  Prelimi- 
nary Survey  to  the  Construction  of  Railroads,  intended  Ex- 
pressly for  the  Young  Engineer,  together  with  Numerous  Valu- 
able Rules  and  Examples.  By  W.  GRISWOLD.  12mo.,  tucks. 

$1  25 
PIUETTIER.— METALLIC  ALLOYS  : 

Being  a  Practical  Guide  to  their  Chemical  and  Physical  Pro- 
perties, their  Preparation,  Composition,  and  Uses.  Translated 
from  the  French  of  A.  GTJETTIER,  Engineer  and  Director  of 
Founderies,  author  of  "  La  Fouderie  en  France,"  etc.  etc.  By 
A.  A.  FESQUET,  Chemist  and  Engineer.  In  one  volume,  12mo. 
(In  press,  shortly  to  be  published.) 

TTATS  AND  FELTING: 

A  Practical  Treatise  on  their  Manufacture.  By  a  Practical 
Hatter.  Illustrated  by  Drawings  of  Machinery,  &c  ,  8vo. 

TTAY.— THE  INTERIOR  DECORATOR  : 

The  Laws  of  Harmonious  Coloring  adapted  to  Interior  Decora- 
tions: with  a  Practical  Treatise  on  House-Painting.  By  D. 
R.  HAY,  House-Painter  and  Decorator.  Illustrated  by  a  Dia- 
gram of  the  Primary,  Secondary,  and  Tertiary  Colors.  12mo. 

$2  25 

TTUGHES.— AMERICAN    MILLER    AND    MILLWRIGHT'S    AS- 

11     SISTANT : 

By  WM.  CARTER  HUGHES.  A  new  edition.  In  one  volume, 
12mo.  .  $i  50 


HENRY  CAREY  BAIRD'S  CATALOGUE.  13 


TTUNT.— THE  PRACTISE  OF  PHOTOGRAPHY. 

I3y  ROBERT  HUNT,  Vice-President  of  the  Photographic  Society, 
London,  with  numerous  illustrations.  12mo.,  cloth  .  75 

JJURST.— A  HAND-BOOK  FOR  ARCHITECTURAL  SURVEYORS  : 
Comprising  Formulae  useful  in  Designing  Builder's  work,  Table 
of  Weights,  of  the  materials  used  in  Building,  Memoranda 
connected  with  Builders'  work,  Mensuration,  the  Practice  of 
Builders'  Measurement,  Contracts  of  Labor,  Valuation  of  Pro- 
perty, Summary  of  the  Practice  in  Dilapidation,  etc.  etc.  By 
J.  F.  HURST,  C.  E.  2d  edition,  pocket-book  form,  full  bound 

§2  50 

JERVIS— RAILWAY  PROPERTY : 

A  Treatise  on  the  Construction  and  Management  of  Railways  ; 
designed  to  afford  useful  knowledge,  in  the  popular  style,  to  the 
holders  of  this  class  of  property ;  as  well  as  Railway  Mana- 
gers, Officers,  and  Agents.  By  JOHN  B.  JERVIS,  late  Chief 
Engineer  of  the  Hudson  River  Railroad,  Croton  Aqueduct,  &c. 
One  vol.  12mo.,  cloth $2  00 

JOHNSON.— A  REPORT  TO  THE  NAVY  DEPARTMENT  OF  THE 
U      UNITED  STATES  ON  AMERICAN  ^JOALS : 

Applicable  to  Steam  Navigation  and  to  other  purposes.  By 
WALTER  R.  JOHNSON.  With  numerous  illustrations.  GOT  pp. 
8vo.,  half  morocco $6  00 

JOHNSON— THE  COAL  TRADE  OF  BRITISH  AMERICA : 

With  Researches  on  the  Characters  and  Practical  Values  of 

.American  and  Foreign  Coals.     By  WALTER  R.  JOHNSON,  Civil 

and  Mining  Engineer  and  Chemist.     8vo.  .         .         .     $2  00 

JOHNSTON.— INSTRUCTIONS  FOR  THE  ANALYSIS   OF  SOILS, 
"      LIMESTONES,  AND  MANURES. 

By  J.  W.  F.  JOHNSTON.     12mo 38 

T7-EENE.— A  HAND-BOOK  OF  PRACTICAL  GAUGING, 

For  the  Use  of  Beginners,  to  which  is  added  A  Chapter  on  Dis- 
tillation, describing  the  process  in  operation  at  the  Custom 
House  for  ascertaining  the  strength  of  wines.  By  JAMES  B. 

KEENE,  of  H.  M.  Customs.     8vo $  I  25 

tfENTISH.— A  TREATISE  ON  A  BOX  OF  INSTRUMENTS, 

r% 

And  the  Slide  Rule ;  with  the  Theory  of  Trigonometry  and  Lo- 
garithms, including  Practical  Geometry,  Surveying,  Measur- 
ing of  Timber,  Cask  and  Malt  Gauging,  Heights,  and  Distances. 
By  THOMAS  KENTISH.  In  one  volume.  12mo.  .  $1  25 


14  HENRY  CARET  BAIRD'S  CATALOGUE. 


TTOBELL.— ERNL—  MINERALOGY  SIMPLIFIED  : 

A  short  method  of  Determining  and  Classifying  Minerals,  by 
means  of  simple  Chemical  Experiments  in  the  Wet  Way. 
Translated  from  the  last  German  Edition  of  F.  VON  KOBELL, 
•with  an  Introduction  to  Blowpipe  Analysis  and  other  addi- 
tions. By  HENRI  EENI,  M.  D.,  Chief  Chemist,  Department  of 
Agriculture,  author  of  "  Coal  Oil  and  Petroleum."  In  one 
volume,  12mo.  .  •-...•  .  .  .  .  $2  50 

TAFFINEUR.— A  PRACTICAL  GUIDE  TO  HYDRAULICS  FOR 

•"     TOWN  AND  COUNTRY; 

Or  a  Complete  Treatise  on  the  Building  of  Conduits  for  Water 
for  Cities,  Towns,  Farms,  Country  Residences,  Workshops,  etc. 
Comprising  the  means  necessary  for  obtaining  at  all  times 
abundant  supplies  of  Drinkable  Water.  Translated  from 
the  French  of  M.  JULES  LAFFINEUR,  C.  E.  Illustrated.  (In 
press.) 

T  AFFINEUR.— A  TREATISE  ON  THE  CONSTRUCTION  OF  WA- 
*-*     TER- WHEELS : 

Containing  the  variou*  Systems  in  use  with  Practical  Informa- 
tion on  the  Dimensions  necessary  for  Shafts,  Journals,  Arms, 
etc.,  of  Water-wheels,  etc.  etc.  Translated  from  the  French 
of  M.  JULES  LAFFINEUR,  C.  E.  Illustrated  by  numerous 
plates.  (In  press.) 

T  ANDRIN.— A  TREATISE  ON  STEEL : 

Comprising  the  Theory,  Metallurgy,  Practical  Working,  Pro- 
perties, and  Use.  Translated  from  the  French  of  II.  C.  LAN- 
DRIN,  Jr.,  C.  E.  By  A.  A.  FESQUET,  Chemist  and  Engineer. 
Illustrated.  12mo.  (In  press.) 

TARKIN.— THE  PRACTICAL  BRASS  AND   IRON  FOUNDER'S 
*-*     GUIDE : 

A  Concise  Treatise  on  Brass  Founding,  Moulding,  the  Metals 
and  their  Alloys,  etc. ;  to  which  are  added  Recent  Improve- 
ments in  the  Manufacture  of  Iron,  Steel  by  the  Bessemer  Pro- 
cess, etc.  etc.  By  JAMES  LARKIN,  late  Conductor  of  the  Brass 
Foundry  Department  in  Reany,  Neafie  &  Co.'s  Penn  Works, 
Philadelphia.  Fifth  edition,  revised,  •with  Extensive  addi- 
tions. In  one  volume,  12mo $2  25 


HENRY  CAREY  BAIRD'S  CATALOGUE.        15 

T  EAVITT.— FACTS  ABOUT  PEAT  AS  AN  ARTICLE  OF  FUEL : 
With  Remarks  upon  its  Origin  and  Composition,  the  Localities 
in  which  it  is  found,  the  Methods  of  Preparation  and  Manu- 
facture, and  the  various  Uses  to  which  it  is  applicable ;  toge- 
ther with  many  other  matters  of  Practical  and  Scientific  Inte- 
rest. To  which  is  added  a  chapter  on  the  Utilization  of  Coal 
Dust  with  Peat  for  the  Production  of  an  Excellent  Fuel  at 
Moderate  Cost,  especially  adapted  for  Steam  Service.  By  II. 
T.  LEAVITT.  Third  edition.  12mo.  .  .  .  $1  75 

T  EEOUX  —  A  PEACTICAL  TEEATISE   OTT  WOOLS  AND  WOE- 
•*-*     STEDS : 

Translated  from  the  French  of  CHARLES  LEROUX,  Mechanical 
Engineer,  and  Superintendent  of  a  Spinning  Mill.  Illustrated 
by  12  large  plates  and  34  engravings.  In  one  volume  8vo. 
(In  press,  shortly  to  be  published.") 

TESLIE  (MISS).— COMPLETE  COOKEEY  : 

Directions  for  Cookery  in  its  Various  Branches.      By  Miss 
LESLIE.     58th  thousand.    Thoroughly  revised,  with  the  addi- 
tion of  New  Receipts.     In  1  vol.  12mo.,  cloth    .         .     $1  25 
T  ESLIE  (MISS).  LADIES'  HOUSE  BOOK  : 

a  Manual  of  Domestic  Economy.  20th  revised  edition.  12mo., 
cloth  .  . $1  25 

TESLIE    (MISS).— TWO    HUNDEED    EECEIPTS    IN   FEENCH 
COOKEEY. 
12mo 50 

T  IEBEB.— ASSAYEE'S  GUIDE: 

Or,  Practical  Directions  to  Assayers,  Miners,  and  Smelters,  for 
the  Tests  and  Assays,  by  Heat  and  by  Wet  Processes,  for  the 
Ores  of  all  the  principal  Metals,  of  Gold  and  Silver  Coins  and 
Alloys,  and  of  Coal,  etc.  By  OSCAR  M.  LIBBER.  12mo.,  cloth 

$1  25 

T  OVE.— THE  AET  OF  DYEING,  CLEANING,  SCOUEING,  AND 

71     FINISHING : 

On  the  most  approved  English  and  French  methods ;  being 
Practical  Instructions  in  Dyeing  Silks,  Woollens,  and  Cottons, 
Feathers,  Chips,  Straw,  etc.;  Scouring  and  Cleaning  Bed  and 
Window  Curtains,  Carpets,  Rugs,  etc.;  French  and  English 
Cleaning,  any  Color  or  Fabric  of  Silk,  Satin,  or  Damask.  By 
THOMAS  LOVE,  a  Working  Dyer  and  Scourer.  In  1  vol.  12mo. 

$3  00 


16  HENRY  CARET  BAIRD'S  CATALOGUE. 

T\yr  YIN  AND  BROWN.— QUESTIONS  ON  SUBJECTS  CONNECTED 

1V1  WITH  THE  MARINE  STEA  ML -ENGINE: 

And  Examination  Papers  ;  with  Hints  for  their  Solution.  By 
THOMAS  J.  MAIN,  Professor  of  Mathematics,  Royal  Naval  Col- 
lege, and  THOMAS  BROWN,  Chief  Engineer,  R.  N.  12mo.,  cloth 

$1  50 

•R/TAJN  AND  BROWN  —THE  INDICATOR  AND  DYNAMOMETER; 
With  their  Practical  Applications  to  the  Steam-Eugine.  By 
THOMAS  J.  MAIN,  M.  A.  F.  R.,  Ass't  Prof.  Royal  Naval  College, 
Portsmouth,  and  THOMAS  BROWN,  Assoc.  Inst.  C.  E.,  Chief  En- 
gineer, R.  N.,  attached  to  the  R.  N.  College.  Illustrated. 
From  the  Fourth  London  Edition.  8vo.  .  .  .  $1  50 

TUT  .UN  AND  BROWN.— THE  MARINE  STEAM-ENGINE. 

By  THOMAS  J.  MAIN,  F.  R.  Ass't  S.  Mathematical  Professor  at 
Royal  Naval  College,  and  THOMAS  BROWN,  Assoc.  Inst.  C.  E. 
Chief  Engineer,  R.  N.  Attached  to  the  Royal  Naval  College. 
Authors  of  "  Questions  connected  with  the  Marine  Steam-En- 
gine,"  and  the  "  Indicator  and  Dynamometer."  With  nume- 
rous Illustrations.  In  one  volume,  8vo.  .  .  .  $5  00 

TUTAKINS.— A  MANUAL  OF  METALLURGY : 

More  particularly  of  the  Precious  Metals:  including  the  Meth- 
ods of  Assaying  them.  Illustrated  by  upwards  of  50  Engrav- 
ings. By  GEORGE  HOGARTH  MAKINS,  M.  R.  C.  S.,  F.  C.  S.,  one 
of  the  Assayers  to  the  Bank  of  England,  Assayer  to  the  Anglo- 
Mexican  Mints,  and  Lecturer  upon  Metallurgy  at  the  Dental 
Hospital,  London.  In  one  volume,  12mo.  .  .  $3  50 

TV/TARTIN— SCEEW-CUTTING  TABLES,  FOR  THE  USE  OF  ME- 

1V1   CHANICAL  ENGINEERS : 

Showing  the  Proper  Arrangement  of  Wheels  for  Cutting  the 
Threads  of  Screws  of  any  required  Pitch ;  with  a  Table  for 
Making  the  Universal  Gas-Pipe  Thread  and  Taps.  By  W.  A. 
MARTIN,  Engineer.  8vo.  .....  50 

TUTILES.— A  PLAIN  TREATISE  ON  HORSE-SHOEING. 

With  illustrations.  By  WILLIAM  MILES,  author  of  "The 
Horse's  Foot,"  .  .  .  . '  .  .  $1  00 

MDLESWORTH.  POCKET-BOOK  OF  USEFUL  FORMULAE  AND 
MEMORANDA  FOR  CIVIL  AND  MECHANICAL  ENGI- 
NEERS. 

By  GniLFORD  L.  MOLESWORTH,  Member  of  the  Institution  of 
Civil  Engineers,  Chief  Resident  Engineer  of  the  Ceylon  Rail- 
way. Second  American,  from  the  Tenth  London  Edition.  In 
one  volume,  full  bound  in  pocket-book  form  .  .  $2  00 


HENRY  CAREY  BAIRD'S  CATALOGUE.  17 

TV/TOORE.—  THE  INVENTOR'S  GUIDE: 

Patent  Office  and  Patent  Laws;  or,  a  Guide  to  Inventors,  and 
a  Book  of  Reference  for  Judges,  Lawyers,  Magistrates,  and 
others.     By  J.  G.  MOORE.     12mo.,  cloth        .        .         $125 
TWpREAU.—  PRACTICAL  GUIDE  FOR  THE  JEWELLER, 

In  the  Application  of  Harmony  of  Colors  in  the  Arrangement 
of  Precious  Stones,  Gold,  etc.,  from  the  French  of  M.  L.  Mo- 
REAU,  Jeweller  and  Designer.  Illustrated.  (In  press.) 

•JJ1PIER.—  CHEMISTRY  APPLIED  TO  DYEING. 

By  JAMES  NAPIER,  F.  C.  S.  A  new  and  revised  edition, 
brought  down  to  the  present  condition  of  the  Art.  Illustrated. 
(In  press.) 

1\T<i.PIER.—  A  MANUAL  OF  DYEING  RECEIPTS  FOR  GENERAL 
1N     USE. 

By  JAMES  NAPIER,  F.  C  S.  With  Numerous  Patterns  of  Dyed 
Cloth  and  Silk.  Second  edition,  revised  and  enlarged.  12mo. 

$3  75 
•RTAPIER.—  MANUAL  OF  ELECTRO-METALLURGY: 

Including  the  Application  of  the  Art  to  Manufacturing  Pro- 
cesses. By  JAMES  NAPIER.  Fourth  American,  from  the 
Fourth  London  edition,  revised  and  enlarged.  Illustrated  by 
engravings.  In  one  volume,  8vo.  .  .  .  $2  00 

TfTEWBERY.  —  GLEANINGS    FROM    ORNAMENTAL    ART    OF 
•^     EVERY  STYLE; 

Drawn  from  Examples  in  the  British,  South  Kensington,  In- 
dian, Crystal  Palace,  and  other  Museums,  the  Exhibitions  of 
1851  and  1862,  and  the  best  English  and  Foreign  works.  In 
a  series  of  one  hundred  exquisitely  drawn  Plates,  containing 
many  hundred  examples.  By  ROBERT  NEWBERY.  4to.  $1500 

TVTICHOLSON.—  A  MANUAL  OF  THE  ART  OF  BOOK-BINDING  : 

Containing  full  instructions  in  the  different  Branches  of  For- 
warding, Gilding,  and  Finishing.  Also,  the  Art  of  Mai'bling 
Book-edges  and  Paper.  By  JAMES  B.  NICHOLSON.  Illus- 
trated. 12mo.,  cloth  ......  $2  25 


.—  A  HAND-BOOK  FOR  LOCOMOTIVE  ENGINEERS  AND 
MACHINISTS  : 

Comprising  the  Proportions  and  Calculations  for  Constructing 
Locomotives  ;  Manner  of  Setting  Valves  ;  Tables  of  Squares, 
Cubes,  Areas,  etc.  etc.  By  SEPTIMUS  Nonius,  Civil  and  Me- 
chanical Engineer.  New  edition.  Illustrated,  12mo.,  cloth 

$2  00 


18  HENRY  CAREY  BAIRD'S  CATALOGUE. 

•MTSTROM.  —  ON    TECHNOLOGICAL    EDUCATION    AND    THE 
CONSTRUCTION  OF  SHIPS  AND  SCREW  PROPELLERS : 
For  Naval  and  Marine  Engineers.    By  JOHN  W.  NYSTKOM,  late 
Acting  Chief  Engineer  U.  S.  N.  'Second  edition,  revised  with 
additional  matter.     Illustrated  by  seven  engravings.    12mo. 

$2  50 

fVNEILL.— CHEMISTRY  OF  CALICO  PRINTING,  DYEING,  AND 

U     BLEACHING : 

Including  Silken,  Woollen,  and  Mixed  Goods ;  Practical  and 
Theoretical.  By  CHARLES  O'NEILL.  (In  press.) 

(VNEILL.— A  DICTIONARY  OF  CALICO  PRINTING  AND  DYE- 

U     ING: 

Containing  a  Brief  Account  of  all  the  Substances  and  Processes 
in  Use  in  the  Arts  of  Printing  and  Dyeing  Textile  Fabrics;  with 
Practical  Receipts  and  Scientific  Information.  By  CHARLES 
O'NEILL,  Analytical  Chemist,  Fellow  of  the  Chemical  Society 
of  London,  etc.  etc.  Author  of  "  Chemistry  of  Calico  Print- 
ing and  Dyeing."  8vo.  (In  press.) 

rjVERMAN— OSBORN.— THE  MANUFACTURE  OF  IRON  IN  ALL 
^     ITS  BRANCHES: 

Including  a  Practical  Description  of  the  various  Fuels  and 
their  Values,  the  Nature,  Determination  and  Preparation  of 
the  Ore,  the  Erection  and  Management  of  Blast  and  other  Fur- 
naces, the  characteristic  results  of  Working  by  Charcoal, 
Coke,  or  Anthracite,  the  Conversion  of  the  Crude  into  the  va- 
rious kinds  of  Wrought  Iron,  and  the  Methods  adapted  to  this 
end.  Also,  a  Description  of  Forge  Hammers,  Rolling  Mills, 
Blast  Engines,  &c.  &c.  To  which  is  added  an  Essay  OQ  the 
Manufacture  of  Steel.  By  FREDERICK  OVERMAN,  Mining  En- 
gineer. The  whole  thoroughly  revised  and  enlarged,  adapted 
to  the  latest  Improvements  and  Discoveries,  atid  the  particular 
type  of  American  Methods  of  Manufacture.  With  various 
new  engravings  illustrating  the  whole  subject.  By  H.  S.  OS- 
BORN,  LL.  D.  Professor  of  Mining  and  Metallurgy  in  Lafay- 
ette College.  In  one  volume,  8vo.  (In  press.)  .  $10  00 

p A1NTER,  GILDER,  AND  VARNISHER'S  COMPANION : 

Containing  Rules  and.  Regulations  in  everything  relating  to 
the  Arts  of  Painting,  Gilding,  Varnishing,  and  Glass  Staining, 
with  numerous  useful  and  valuable  Receipts;  Tests  for  the 
Detection  of  Adulterations  in  Oils  and  Colors,  and  a  statement 
of  the  Diseases  and  Accidents  to  which  Painters,  Gilders,  and 


HENRY  CAREY  BAIRD'S  CATALOGUE.  19 

Varnishers  are  particularly  liable,  with  the  simplest  methods 
of  Prevention  and  Remedy.  With  Directions  for  Graining. 
Marbling,  Sign  Writing,  and  Gilding  on  Glass.  To  which  are 
added  COMPLETE  INSTRUCTIONS  FOE  COACH  PAINTING  AND  VAR- 
NISHINO.  12mo.,  cloth  .  ...  $1  50 

P.YLLETT.— THE  MILLER'S,   MILLWRIGHT'S,   AND  ENGI- 

r     NEER'S  GUIDE. 

By  HENRY  PALLETT.     Illustrated.    In  one  vol.  12mo.     $3  00 
pERKINS.— GAS  AND  VENTILATION. 

Practical  Treatise  on  Gas  and  Ventilation.  With  Special  Re- 
lation  to  Illuminating,  Heating,  and  Cooking  by  Gas.  Includ- 
ing Scientific  Helps  to  Engineer-students  and  others.  With 
illustrated  Diagrams.  By  E.  E.  PERKINS.  12mo.,  cioth  §1  25 

pERKINS   AND   STOWE.— A   NEW   GUIDE  TO   THE    SHEET- 
r     IRON  AND  BOILER  PLATE  ROLLER : 

Containing  a  Series  of  Tables  showing  the  Weight  of  Slabs  and 
Piles  to  Produce  Boiler  Plates,  and  of  the  Weight  of  Piles  and 
the  Sizes  of  Bars  to  produce  Sheet-iron;  the  Thickness  of  the 
Bar  Gauge  in  Decimals  ;  the  Weight  per  foot,  and  the  Thick- 
ness on  the  Bar  or  Wire  Gauge  of  the  fractional  parts  of  an 
inch ;  the  Weight  per  sheet,  and  the  Thickness  on  the  Wire 
Gauge  of  Sheet- iron  of  various  dimensions  to  weigh  112  Ibs. 
per  bundle ;  and  the  conversion  of  Short  Weight  into  Long 
Weight,  and  Long  Weight  into  Short.  Estimated  and  collected 
by  G.  H.  PERKINS  and  J.  G.  STOWE  .  .  .  .  $2  50 

pZILLIPS  AND  DARLINGTON.— RECORDS  OF  MINING  AND 
r      METALLURGY : 

Or  Facts  and  Memoranda  for  the  use  of  the  Mine  Agent  and 
Smelter.  By  J.  ARTHUR  PHILLIPS,  Mining  Engineer,  Graduate 
of  the  Imperial  School  of  Mines,  France,  etc.,  and  JOHN  DAR- 
LINGTON. Illustrated  by  numerous  engravings.  In  one  vol- 
ume, 12mo $2  00 

p3,ADAL,    MALEPEYRE,    AND    DUSSAUCE.  —  A    COMPLETE 
*      TREATISE  ON  PERFUMERY  : 

Containing  notices  of  the  Raw  Material  used  in  the  Art,  and  the 
Best  Formulae.  According  to  the  most  approved  Methods  fol- 
lowed in  France,  England,  and  the  United  States.  By  M. 
P.  PRADAL,  Perfumer  Chemist,  and  M.  F.  MALEPEYRE.  Trans- 
lated from  the  French,  with  extensive  additions,  by  Professor 
II.  DCSSAUCE.  8vo.  .  .  .  .  .  .  .  $10  00 


20  HENRY  CAREY  CAIRO'S  CATALOGUE. 

pSOTEAUX  — PEACTICAL  GUIDE   FOR  THE   MANUFACTURE 

r      OF  PAPER  AND  BOARDS. 

By  A.  PEOTEAUX,  Civil  Engineer,  and  Graduate  of  the  School 
of  Arts  and  Manufactures,  Director  of  Thiers's  Paper  Mill, 
'Puy-de-Dome.  With  additions,  by  L.  S.  LE  XORMAXI). 
Translated  from  the  French,  with  Notes,  by  HORATIO  PAIXK, 
A.  B.,  M.  D.  To  which  is  added  a  Chapter  on  the  Manufac- 
ture of  Paper  from  Wood  in  the  United  States,  by  HENRY  T. 
BROWN,  of  the  "American  Artisan."  Illustrated  by  six  plates, 
containing  Drawings  of  Raw  Materials,  Machinery,  Plans  of 
Paper-Mills,  etc.  etc.  8vo $3  00 

•DEGNAULT.— ELEMENIS  OF  CHEMISTRY. 

By  M.  V.  HEGNACLT.  Translated  from  the  French  by  T. 
FORREST  BETTON,  M.D.,  and  edited,  with  notes,  by  JAMES  C. 
BOOTH,  Melter  and  Refiner  U.  S.  Mint,  and  WM.  L.  FABER, 
Metallurgist  and  Mining  Engineer.  Illustrated  by  nearly  700 
•  wood  engravings.  Comprising  nearly  1500  pages.  In  tvro 
volumes,  8vo.,  cloth §10  00 

OELLERS— THE  COLOR  MIXER : 

Containing  nearly  Four  Hundred  Receipts  for  Colors,  Pastes, 
Acids,  Pulps,  Blue  Vats,  fciquors,  etc.  etc.,  for  Cotton  and 
Woollen  Goods:  including  the  celebrated  Barrow  Delaine  Co- 
lors. By  JOHN  SELLERS,  an  experienced  Practical  AVorkman. 

In  one  volume,  12mo. $2  50 

OHUNK— A   PRACTICAL   TREATISE  ON  RAILWAY  CURVES 
^    -AND  LOCATION,  FOR  YOUNG  ENGINEERS. 

By  WM.  F.  SHUXE,  Civil  Engineer.     12mo.         .         .     $1  50 

qXEATON— -BUILDER'S  POCKET  COMPANION: 

Containing  the  Elements  of  Building,  Surveying,  and  Archi- 
tecture ;  with  Practical  Rules  and  Instructions  connected  with 
the  subject.  By  A.  C.  SMEATON,  Civil  Engineer,  etc.  In 
one  volume,  12mo .  .  $1  25 

a-jJITH  — THE  DYER'S  INSTRUCTOR: 

Comprising  Practical  Instructions  in  the  Art  of  Dyeing  Silk, 
Cotton,  Wool,  and  Worsted,  and  Woollen  Goods:  containing 
nearly  800  Receipts.  To  which  is  added  a  Treatise  on  the  Art 
of  Padding ;  and  the  Printing  of  Silk  Warps,  Skeins,  and 
Handkerchiefs,  and  the  various  Mordants  and  Colors  for  the 
different  styles  of  such  work.  By  DAVID  SMITH,  Pattern 
Dyer.  12mo.,  cloth $3  00 


HENRY  CAREY  BAIRD'S  CATALOGUE.  21 

.— PARKS  AND  PLEASURE  GROUNDS  : 

Or  Practical  Notes  on  Country  Residences,  Villas,  Public 
Parks,  and  Gardens.  By  CHARLES  II.  J.  SJIITIJ,  Landscape 
Gardener  and  Garden  Architect,  etc.  etc.  12mo.  .  $2  25 

OTOKES.— CABINET-MAKER'S  AND  UPHOLSTERER'S  COMPA- 
°     NION : 

Comprising  the  Rudiments  and  Principles  of  Cabinet-making 
and  Upholstery,  with  Familiar  Instructions,  Illustrated  by  Ex- 
amples for  attaining  a  Proficiency  in  the  Art  of  Drawing,  MS 
applicable  to  Cabinet-work;  The  Processes  of  Veneering,  In- 
laying, and  Buhl-work;  the  Art  of  Dyeing  and  Staining  Wood, 
Bone,  Tortoise  Shell,  etc.  Directions  for  .Lackering,  Japan- 
ning, and  Varnishing ;  to  make  French  Polish ;  to  prepare  the 
Best  Glues,  Cements,  and  Compositions,  and  a  number  of  Re- 
ceipts particularly  for  workmen  generally.  By  J.  STOKES.  In 
one  vol.  12mo.  With  illustratipns  .  .  .  .  $1  25 

STRENGTH  AND  OTHER  PROPERTIES  OF  METALS. 

Reports  of  Experiments  on  the  Strength  and  other  Proper- 
ties of  Metals  for  Cannon.  With  a  Description  of  the  Machines 
for  Testing  Metals,  and  of  the  Classification  of  Cannon  in  ser- 
vice. By  Officers  of  the  Ordnance  Department  U.  S.  Army 
By  authority  of  the  Secretary  of  War.  Illustrated  by  25  large 
steel  plates.  In  1  vol.  quarto  .  .  .  .  .  $10  00 

rpABLES  SHOWING  THE  WEIGHT  OF  ROUND,  SQUARE,  AND 
1     FLAT   BAR  IRON,  STEEL,  ETC., 

By  Measurement.     Cloth G3 

rnAYLOR.— STATISTICS  OF  COAL  : 

Including  Mineral  Bituminous  Substances  employed  in  Arts 
and  Manufactures;  with  their  Geographical,  Geological,  and 
Commercial  Distribution  and  amount  of  Production  and  Con- 
sumption on  the  American  Continent.  With  Incidental  Sta- 
tistics of  the  Iron  Manufacture.  By  R.  C.  TAYLOR.  Second 
edition,  revised  by  S.  S.  HALDEMAN.  Illustrated  by  five  Maps 
and  many  wood  engravings.  8vo.,  cloth  .  .  .  $3  00 

rP3MPLETON.— THE    PRACTICAL   EXAMINATOR   ON   STEAM 

-1     AND  THE  STEAM-ENGINE  : 

.With  Instructive  References  relative  thereto,  for  the  Use  of 
Engineers,  Students,  and  others.  By  WM.  TEMPLETON,  Engi- 
neer. 12mo .  .  $1  25 


22  HENRY  CAREY  BAIRD'S  CATALOGUE. 

mHOMAS.— THE  MODEBN  PRACTICE  OF  PHOTOGRAPHY. 

By  K.  W.  THOMAS,  F.  C.  S.     8vo.,  cloth    ...  75 

rpHOMSON.— FREIGHT  CHARGES  CALCULATOR. 

By  ANDREW  THOMSON,  Freight  Agent         .         .         .     $1  25 

rPURNBULL.— THE  ELECTRO-MAGNEIIC  TELESRAPH : 

With  an  Historical  Account  of  its  Rise,  Progress,  and  Present 
Condition.  Also,  Practical  Suggestions  in  regard  to  Insula- 
tion and  Protection  from  the  effects  of  Lightning.  Together 
with  an  Appendix,  containing  several  important  Telegraphic 
Devices  and  Laws.  By  LAWRENCE  TNRNBULL,  M.  D.,  Lectu- 
rer on  Technical  Chemistry  at  the  Franklin  Institute.  Revised 
and  improved.  Illustrated.  8vo.  .  .  .  $J  00 

TURNER'S  (THE)  COMPANION: 

Containing  Instructions  in  Concentric,  Elliptic,  and  Eccentric 
Turning;  also  various  Plates  of  Chucks,  Tools,  and  Instru- 
ments ;  and  Directions  for  using  the  Eccentric  Cutter,  Drill, 
Vertical  Cutter,  and  Circular  Rest ;  with  Patterns  and  Instruc- 
tions for  working  them.  A  new  edition  in  one  vol.  12mo. 

$1  50 

TTLRICH— DUSSAUCE.— A  COMPLETE  TREATISE  ON  THE  ART 
U     OF  DYEING  COTTON  AND  WOOL: 

As  practised  in  Paris,  Rouen,  Mulhausen,  and  Germany. 
From  the  French  of  M.  Louis  ULRICH,  a  Practical  Dyer  in 
the  principal  Manufactories  of  Paris,  Rouen,  Mulhausen,  etc. 
etc. ;  to  which  are  added  the  most  important  Receipts  for  Dye- 
ing Wool,  as  practised  in  the  Manufacture  Impcriale  des  Go- 
belins, Paris.  By  Professor  H.  DUSSAUCE.  12mo.  $3  00 

TjaJBIN— BRULL.  —  A    PRACTICAL    GUIDE    FOE    PUDDLING 

U      IRON  AND  STEEL. 

By  ED.  URBIN,  Engineer  of  Arts  and  Manufactures.  A  Prize 
Essay  read  before  the  Association  of  Engineers,  Graduate  of 
the  School  of  Mines,  of  Liege,  Belgium,  at  the  Meeting  of 
18G5 — 6.  To  which  is  added  a  COMPARISON  OF  THE  RESISTING 
PROPERTIES  OF  IRCTS  AND  STEEL.  By  A.  BRULL.  Translated 
from  the  French  by  A.  A.  FESQUET,  Chemist  and  Engineer.  In 
one  volume,  8vo.  . $1  00 

TOTATSON.— A  MANUAL  OF  THE  HAND-LATHE. 

By  EGBERT  P.  WATSON,  Late  of  the  "Scientific  Ameriean," 
Author  of  "  Modern  Practice  of  American  Machinists  and 
Engineers."  In  one  volume,  12mo.  (In  press.) 


HENRY  CAREY  BAIRD'S  CATALOGUE.  23 


T70-ATSON.— THE   MODERN    PRACTICE    OF    AMERICAN    MA- 
*    CHINISTS  AND  ENGINEERS  : 

Including  the  Construction,  Application,  and  Use  of  Drills, 
Lathe  Tools,  Cutters  for  Boring  Cylinders,  and  Hollow  Work 
Generally,  with  the  most  Economical  Speed  of  the  same,  the 
Results  verified  by  Actual  Practice  at  the  Lathe,  the  Vice,  and 
on  the  Floor.  Together  with  Workshop  management,  Economy 
of  Manufacture,  the  Steam-Engine,  Boilers,  Gears,  Belting,  etc. 
etc.  By  EGBERT  P.  WATSON,  late  of  the  "  Scientific  American." 
Illustrated  by  eighty-six  engravings.  12mo.  .  .  $2  50 

TXTATSON.— THE  THEORY  AND  PRACTICE   OF  THE  ART  OF 
"    WEAVING  BY  HAND  AND  POWER : 

With  Calculations  and  Tables  for  the  use  of  those  connected 
with  the  Trade.  By  JOHN  WATSON,  Manufacturer  and  Prac- 
tical Machine  Maker.  Illustrated  by  large  drawings  of  the 
best  Power-Looms.  8vo.  .  .  .  .  .  $7  50 

•nTTEATHERLY.— TREATISE  ON  THE  AST   OF  BOILING    STJ- 
""    GAR,  .  CRYSTALLIZING,    LOZENGE-MAKING,    COMFITS, 
GUM  GOODS, 

And  other  processes  for  Confectionery,  &c.  In  which  are  ex- 
plained, in  an  easy  and  familiar  manner,  the  various  Methods 
of  Manufacturing  every  description  of  Raw  and  Refined  sugar 
Goods,  as  sold  by  Confectioners  and  others  .  .  $2  00 

TTT7TLL.— TABLES  FOR  QUALITATIVE  CHEMICAL  ANALYSIS. 
By  Prof.  HEINRICH  WILL,  of  Giessen,  Germany.  Seventh  edi- 
tion. Translated  by  CHARLES  F.  HIMES,  Ph.  D.,  Professor  of 
Natural  Science,  Dickinson  College,  Carlisle,  Pa.  .  $1  25 

TmLLIAMS.— ON  HEAT  AND  STEAM  : 

Embracing  New  Views   of  Vaporization,    Condensation,   and 

Expansion.     By  CHARLES  WYE  WILLIAMS,  A.  I.  C.  E.     Illus- 

-  trated.    8vo $3  50 


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